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Tweedie distribution - Wikipedia
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class="vector-toc-numb">2</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Additive_exponential_dispersion_models" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Additive_exponential_dispersion_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Additive exponential dispersion models</span> </div> </a> <ul id="toc-Additive_exponential_dispersion_models-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reproductive_exponential_dispersion_models" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reproductive_exponential_dispersion_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Reproductive exponential dispersion models</span> </div> </a> <ul id="toc-Reproductive_exponential_dispersion_models-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scale_invariance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scale_invariance"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Scale invariance</span> </div> </a> <ul id="toc-Scale_invariance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Tweedie_power_variance_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Tweedie_power_variance_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>The Tweedie power variance function</span> </div> </a> <ul id="toc-The_Tweedie_power_variance_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Tweedie_deviance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Tweedie_deviance"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>The Tweedie deviance</span> </div> </a> <ul id="toc-The_Tweedie_deviance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Tweedie_cumulant_generating_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Tweedie_cumulant_generating_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>The Tweedie cumulant generating functions</span> </div> </a> <ul id="toc-The_Tweedie_cumulant_generating_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Tweedie_convergence_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Tweedie_convergence_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>The Tweedie convergence theorem</span> </div> </a> <ul id="toc-The_Tweedie_convergence_theorem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_distributions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Related distributions</span> </div> </a> <ul id="toc-Related_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Occurrence_and_applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Occurrence_and_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Occurrence and applications</span> </div> </a> <button aria-controls="toc-Occurrence_and_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Occurrence and applications subsection</span> </button> <ul id="toc-Occurrence_and_applications-sublist" class="vector-toc-list"> <li id="toc-The_Tweedie_models_and_Taylor’s_power_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Tweedie_models_and_Taylor’s_power_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>The Tweedie models and Taylor’s power law</span> </div> </a> <ul id="toc-The_Tweedie_models_and_Taylor’s_power_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tweedie_convergence_and_1/f_noise" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tweedie_convergence_and_1/f_noise"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Tweedie convergence and 1/<i>f</i> noise</span> </div> </a> <ul id="toc-Tweedie_convergence_and_1/f_noise-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Tweedie_models_and_multifractality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Tweedie_models_and_multifractality"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>The Tweedie models and multifractality</span> </div> </a> <ul id="toc-The_Tweedie_models_and_multifractality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Regional_organ_blood_flow" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Regional_organ_blood_flow"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Regional organ blood flow</span> </div> </a> <ul id="toc-Regional_organ_blood_flow-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cancer_metastasis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cancer_metastasis"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Cancer metastasis</span> </div> </a> <ul id="toc-Cancer_metastasis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Genomic_structure_and_evolution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Genomic_structure_and_evolution"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Genomic structure and evolution</span> </div> </a> <ul id="toc-Genomic_structure_and_evolution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Random_matrix_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Random_matrix_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Random matrix theory</span> </div> </a> <ul id="toc-Random_matrix_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_distribution_of_prime_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_distribution_of_prime_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8</span> <span>The distribution of prime numbers</span> </div> </a> <ul id="toc-The_distribution_of_prime_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.9</span> <span>Other applications</span> </div> </a> <ul id="toc-Other_applications-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Family of probability distributions</div> <p class="mw-empty-elt"> </p><p>In <a href="/wiki/Probability" title="Probability">probability</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, the <b>Tweedie distributions</b> are a family of <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a> which include the purely continuous <a href="/wiki/Normal_distribution" title="Normal distribution">normal</a>, <a href="/wiki/Gamma_distribution" title="Gamma distribution">gamma</a> and <a href="/wiki/Inverse_gaussian_distribution" class="mw-redirect" title="Inverse gaussian distribution">inverse Gaussian</a> distributions, the purely discrete scaled <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a>, and the class of <a href="/wiki/Compound_poisson_distribution#Compound_Poisson_Gamma_distribution" class="mw-redirect" title="Compound poisson distribution">compound Poisson–gamma</a> distributions which have positive mass at zero, but are otherwise continuous.<sup id="cite_ref-t84_1-0" class="reference"><a href="#cite_note-t84-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Tweedie distributions are a special case of <a href="/wiki/Exponential_dispersion_model" title="Exponential dispersion model">exponential dispersion models</a> and are often used as distributions for <a href="/wiki/Generalized_linear_model" title="Generalized linear model">generalized linear models</a>.<sup id="cite_ref-Jørgensen-1997_2-0" class="reference"><a href="#cite_note-Jørgensen-1997-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The Tweedie distributions were named by <a href="/wiki/Bent_J%C3%B8rgensen_(statistician)" title="Bent Jørgensen (statistician)">Bent Jørgensen</a> in <sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> after <a href="/wiki/Maurice_Tweedie" title="Maurice Tweedie">Maurice Tweedie</a>,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> a statistician and medical physicist at the <a href="/wiki/University_of_Liverpool" title="University of Liverpool">University of Liverpool</a>, UK, who presented the first thorough study of these distributions in 1982 when the conference <sup id="cite_ref-t84_1-1" class="reference"><a href="#cite_note-t84-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> was held. Around the same, time Bar-Lev and Enis published about the same topic.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The (reproductive) Tweedie distributions are defined as subfamily of (reproductive) <a href="/wiki/Exponential_dispersion_model" title="Exponential dispersion model">exponential dispersion models</a> (ED), with a special <a href="/wiki/Mean" title="Mean">mean</a>-<a href="/wiki/Variance" title="Variance">variance</a> relationship. A <a href="/wiki/Random_variable" title="Random variable">random variable</a> <i>Y</i> is Tweedie distributed <i>Tw<sub>p</sub>(μ, σ<sup>2</sup>)</i>, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim \mathrm {ED} (\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼<!-- ∼ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">D</mi> </mrow> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim \mathrm {ED} (\mu ,\sigma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2119957a972fe3ee19fb15fa6eb71a11ee1a77c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.86ex; height:3.176ex;" alt="{\displaystyle Y\sim \mathrm {ED} (\mu ,\sigma ^{2})}"></span> with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\operatorname {E} (Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ<!-- μ --></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\operatorname {E} (Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ef4f6eb5c41c59a8780c340b850cc32e10af7f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.666ex; height:2.843ex;" alt="{\displaystyle \mu =\operatorname {E} (Y)}"></span>, positive dispersion parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} (Y)=\sigma ^{2}\,\mu ^{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (Y)=\sigma ^{2}\,\mu ^{p},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab3ccec6ce2078cbf8e9297df19b14f45f372c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.378ex; height:3.176ex;" alt="{\displaystyle \operatorname {Var} (Y)=\sigma ^{2}\,\mu ^{p},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in \mathbf {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a653f84936f4edc0a0fb088b5b57c4a3b66467a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.103ex; height:2.509ex;" alt="{\displaystyle p\in \mathbf {R} }"></span> is called the Tweedie power parameter. The probability distribution <i>P</i><sub><i>θ</i>,<i>σ</i><sup>2</sup></sub> on the <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measurable sets</a> <i>A</i>, is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\theta ,\sigma ^{2}}(Y\in A)=\int _{A}\exp \left({\frac {\theta \cdot z-\kappa _{p}(\theta )}{\sigma ^{2}}}\right)\cdot \nu _{\lambda }\,(dz),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ<!-- θ --></mi> <mo>,</mo> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>∈<!-- ∈ --></mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mi>exp</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>θ<!-- θ --></mi> <mo>⋅<!-- ⋅ --></mo> <mi>z</mi> <mo>−<!-- − --></mo> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⋅<!-- ⋅ --></mo> <msub> <mi>ν<!-- ν --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ<!-- λ --></mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>d</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\theta ,\sigma ^{2}}(Y\in A)=\int _{A}\exp \left({\frac {\theta \cdot z-\kappa _{p}(\theta )}{\sigma ^{2}}}\right)\cdot \nu _{\lambda }\,(dz),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d10cc0e0b62ce1eafba35fb5d77080071c9fb291" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.106ex; height:6.343ex;" alt="{\displaystyle P_{\theta ,\sigma ^{2}}(Y\in A)=\int _{A}\exp \left({\frac {\theta \cdot z-\kappa _{p}(\theta )}{\sigma ^{2}}}\right)\cdot \nu _{\lambda }\,(dz),}"></span> for some σ-finite measure <i>ν<sub>λ</sub></i>. This representation uses the canonical parameter <i>θ</i> of an exponential dispersion model and <a href="/wiki/Cumulant" title="Cumulant">cumulant function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa _{p}(\theta )={\begin{cases}{\frac {\alpha -1}{\alpha }}\left({\frac {\theta }{\alpha -1}}\right)^{\alpha },&{\text{for }}p\neq 1,2\\-\log(-\theta ),&{\text{for }}p=2\\e^{\theta },&{\text{for }}p=1\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>α<!-- α --></mi> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ<!-- θ --></mi> <mrow> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>p</mi> <mo>≠<!-- ≠ --></mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>log</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ<!-- θ --></mi> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{p}(\theta )={\begin{cases}{\frac {\alpha -1}{\alpha }}\left({\frac {\theta }{\alpha -1}}\right)^{\alpha },&{\text{for }}p\neq 1,2\\-\log(-\theta ),&{\text{for }}p=2\\e^{\theta },&{\text{for }}p=1\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d9d9f310926deff87b0662ea585f932f58c51b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:37.213ex; height:10.509ex;" alt="{\displaystyle \kappa _{p}(\theta )={\begin{cases}{\frac {\alpha -1}{\alpha }}\left({\frac {\theta }{\alpha -1}}\right)^{\alpha },&{\text{for }}p\neq 1,2\\-\log(-\theta ),&{\text{for }}p=2\\e^{\theta },&{\text{for }}p=1\end{cases}}}"></span> where we used <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\frac {p-2}{p-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> <mrow> <mi>p</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {p-2}{p-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b66711a0c6acb2ea3f3ecbcf1d8edca4ce714ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.594ex; height:5.843ex;" alt="{\displaystyle \alpha ={\frac {p-2}{p-1}}}"></span>, or equivalently <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\frac {\alpha -2}{\alpha -1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> <mrow> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\frac {\alpha -2}{\alpha -1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72f3e3f6cf9eecfe8d5cac52af9c463fba73454d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:10.684ex; height:5.343ex;" alt="{\displaystyle p={\frac {\alpha -2}{\alpha -1}}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Additive_exponential_dispersion_models">Additive exponential dispersion models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=3" title="Edit section: Additive exponential dispersion models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The models just described are in the reproductive form. An exponential dispersion model has always a dual: the additive form. If <i>Y</i> is reproductive, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=\lambda Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mi>λ<!-- λ --></mi> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=\lambda Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce1c015468695456abc8bd3d2f117f1bd4fe2dd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.907ex; height:2.176ex;" alt="{\displaystyle Z=\lambda Y}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={\frac {1}{\sigma ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ<!-- λ --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {1}{\sigma ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16cde9e87c097336c8f74f40116c3d7bdcdcbd87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.675ex; height:5.509ex;" alt="{\displaystyle \lambda ={\frac {1}{\sigma ^{2}}}}"></span> is in the additive form ED<sup>*</sup>(<i>θ</i>,<i>λ</i>), for Tweedie <i>Tw<sup>*</sup><sub>p</sub>(μ, λ)</i>. Additive models have the property that the distribution of the sum of independent random variables, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{+}=Z_{1}+\cdots +Z_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>=</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{+}=Z_{1}+\cdots +Z_{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d81e65c1a22ccf5eb0c0031810cbe7198f45e901" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.695ex; height:2.509ex;" alt="{\displaystyle Z_{+}=Z_{1}+\cdots +Z_{n},}"></span> for which <i>Z</i><sub><i>i</i></sub> ~ ED<sup>*</sup>(<i>θ</i>,<i>λ</i><sub><i>i</i></sub>) with fixed <i>θ</i> and various <i>λ</i> are members of the family of distributions with the same <i>θ</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{+}\sim \operatorname {ED} ^{*}(\theta ,\lambda _{1}+\cdots +\lambda _{n}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>∼<!-- ∼ --></mo> <msup> <mi>ED</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msup> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo>,</mo> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{+}\sim \operatorname {ED} ^{*}(\theta ,\lambda _{1}+\cdots +\lambda _{n}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a3e4fd786ab700622cb1b541c71301ab1d4b486" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.577ex; height:2.843ex;" alt="{\displaystyle Z_{+}\sim \operatorname {ED} ^{*}(\theta ,\lambda _{1}+\cdots +\lambda _{n}).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Reproductive_exponential_dispersion_models">Reproductive exponential dispersion models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=4" title="Edit section: Reproductive exponential dispersion models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A second class of exponential dispersion models exists designated by the random variable <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=Z/\lambda \sim \operatorname {ED} (\mu ,\sigma ^{2}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ<!-- λ --></mi> <mo>∼<!-- ∼ --></mo> <mi>ED</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=Z/\lambda \sim \operatorname {ED} (\mu ,\sigma ^{2}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76dd64e52a21262bd4a84075111dcad15f381a88" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.803ex; height:3.176ex;" alt="{\displaystyle Y=Z/\lambda \sim \operatorname {ED} (\mu ,\sigma ^{2}),}"></span> where <i>σ</i><sup>2</sup> = 1/<i>λ</i>, known as reproductive exponential dispersion models. They have the property that for <i>n</i> independent random variables <i>Y</i><sub><i>i</i></sub> ~ ED(<i>μ</i>,<i>σ</i><sup>2</sup>/<i>w</i><sub><i>i</i></sub>), with weighting factors <i>w<sub>i</sub></i> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=\sum _{i=1}^{n}w_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=\sum _{i=1}^{n}w_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99852c05d19348dfcf0c3821312aac26d3589364" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.615ex; height:6.843ex;" alt="{\displaystyle w=\sum _{i=1}^{n}w_{i},}"></span> a weighted average of the variables gives, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w^{-1}\sum _{i=1}^{n}w_{i}Y_{i}\sim \operatorname {ED} (\mu ,\sigma ^{2}/w).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∼<!-- ∼ --></mo> <mi>ED</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>w</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w^{-1}\sum _{i=1}^{n}w_{i}Y_{i}\sim \operatorname {ED} (\mu ,\sigma ^{2}/w).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/414e801f4b1b78204bbefdddc7f3a2adf39fdfaf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.3ex; height:6.843ex;" alt="{\displaystyle w^{-1}\sum _{i=1}^{n}w_{i}Y_{i}\sim \operatorname {ED} (\mu ,\sigma ^{2}/w).}"></span> </p><p>For reproductive models the weighted average of independent random variables with fixed <i>μ</i> and <i>σ</i><sup>2</sup> and various values for <i>w<sub>i</sub></i> is a member of the family of distributions with same <i>μ</i> and <i>σ</i><sup>2</sup>. </p><p>The Tweedie exponential dispersion models are both additive and reproductive; we thus have the <i>duality transformation</i> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\mapsto Z=Y/\sigma ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo stretchy="false">↦<!-- ↦ --></mo> <mi>Z</mi> <mo>=</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\mapsto Z=Y/\sigma ^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b61900a26849e6bc53a5327ab3c23e72cebec94e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.134ex; height:3.176ex;" alt="{\displaystyle Y\mapsto Z=Y/\sigma ^{2}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Scale_invariance">Scale invariance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=5" title="Edit section: Scale invariance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A third property of the Tweedie models is that they are <a href="/wiki/Scale_invariance" title="Scale invariance">scale invariant</a>: For a reproductive exponential dispersion model <i>Tw<sub>p</sub>(μ, σ<sup>2</sup>)</i> and any positive constant <i>c</i> we have the property of closure under scale transformation, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\operatorname {Tw} _{p}(\mu ,\sigma ^{2})=\operatorname {Tw} _{p}(c\mu ,c^{2-p}\sigma ^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <msub> <mi>Tw</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>Tw</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>c</mi> <mi>μ<!-- μ --></mi> <mo>,</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−<!-- − --></mo> <mi>p</mi> </mrow> </msup> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\operatorname {Tw} _{p}(\mu ,\sigma ^{2})=\operatorname {Tw} _{p}(c\mu ,c^{2-p}\sigma ^{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7b7cc88e7f0e6ebace9a35b8b912edd47d272d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.403ex; height:3.343ex;" alt="{\displaystyle c\operatorname {Tw} _{p}(\mu ,\sigma ^{2})=\operatorname {Tw} _{p}(c\mu ,c^{2-p}\sigma ^{2}).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="The_Tweedie_power_variance_function">The Tweedie power variance function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=6" title="Edit section: The Tweedie power variance function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To define the <a href="/wiki/Variance_function" title="Variance function">variance function</a> for exponential dispersion models we make use of the mean value mapping, the relationship between the canonical parameter <i>θ</i> and the mean <i>μ</i>. It is defined by the function <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau (\theta )=\kappa ^{\prime }(\theta )=\mu .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ<!-- τ --></mi> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ<!-- μ --></mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau (\theta )=\kappa ^{\prime }(\theta )=\mu .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d790b542bd615c6789a035754b8c5b0e03cc201" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.271ex; height:3.009ex;" alt="{\displaystyle \tau (\theta )=\kappa ^{\prime }(\theta )=\mu .}"></span> with cumulative function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa (\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ<!-- κ --></mi> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa (\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c081e503c265d4ada38440908f8dfc57b3052cee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.239ex; height:2.843ex;" alt="{\displaystyle \kappa (\theta )}"></span>. The <a href="/wiki/Natural_exponential_family" title="Natural exponential family">variance function</a> <i>V</i>(<i>μ</i>) is constructed from the mean value mapping, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\mu )=\tau ^{\prime }[\tau ^{-1}(\mu )].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>τ<!-- τ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> <mo stretchy="false">[</mo> <msup> <mi>τ<!-- τ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(\mu )=\tau ^{\prime }[\tau ^{-1}(\mu )].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8229c9c9dbf9bc3ed29b20d4791324063a4f0da6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.781ex; height:3.176ex;" alt="{\displaystyle V(\mu )=\tau ^{\prime }[\tau ^{-1}(\mu )].}"></span> </p><p>Here the minus exponent in <i>τ</i><sup>−1</sup>(<i>μ</i>) denotes an inverse function rather than a reciprocal. The mean and variance of an additive random variable is then <span class="texhtml">E(<i>Z</i>) = <i>λμ</i></span> and <span class="texhtml">var(<i>Z</i>) = <i>λV</i>(<i>μ</i>)</span>. </p><p>Scale invariance implies that the variance function obeys the relationship <span class="texhtml"><i>V</i>(<i>μ</i>) = <i>μ</i><sup> <i>p</i></sup></span>.<sup id="cite_ref-Jørgensen-1997_2-1" class="reference"><a href="#cite_note-Jørgensen-1997-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="The_Tweedie_deviance">The Tweedie deviance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=7" title="Edit section: The Tweedie deviance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The unit <a href="/wiki/Deviance_(statistics)" title="Deviance (statistics)">deviance</a> of a reproductive Tweedie distribution is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(y,\mu )={\begin{cases}(y-\mu )^{2},&{\text{for }}p=0\\2(y\log(y/\mu )+\mu -y),&{\text{for }}p=1\\2(\log(\mu /y)+y/\mu -1),&{\text{for }}p=2\\2\left({\frac {\max(y,0)^{2-p}}{(1-p)(2-p)}}-{\frac {y\mu ^{1-p}}{1-p}}+{\frac {\mu ^{2-p}}{2-p}}\right),&{\text{else}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>μ<!-- μ --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−<!-- − --></mo> <mi>μ<!-- μ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mo stretchy="false">(</mo> <mi>y</mi> <mi>log</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>μ<!-- μ --></mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>μ<!-- μ --></mi> <mo>−<!-- − --></mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>μ<!-- μ --></mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mn>0</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−<!-- − --></mo> <mi>p</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−<!-- − --></mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>y</mi> <msup> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−<!-- − --></mo> <mi>p</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−<!-- − --></mo> <mi>p</mi> </mrow> </msup> <mrow> <mn>2</mn> <mo>−<!-- − --></mo> <mi>p</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>else</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(y,\mu )={\begin{cases}(y-\mu )^{2},&{\text{for }}p=0\\2(y\log(y/\mu )+\mu -y),&{\text{for }}p=1\\2(\log(\mu /y)+y/\mu -1),&{\text{for }}p=2\\2\left({\frac {\max(y,0)^{2-p}}{(1-p)(2-p)}}-{\frac {y\mu ^{1-p}}{1-p}}+{\frac {\mu ^{2-p}}{2-p}}\right),&{\text{else}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0abcda4c1352aea3dcb5eeff668ab4b3977a9be7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; margin-top: -0.237ex; width:54.702ex; height:14.843ex;" alt="{\displaystyle d(y,\mu )={\begin{cases}(y-\mu )^{2},&{\text{for }}p=0\\2(y\log(y/\mu )+\mu -y),&{\text{for }}p=1\\2(\log(\mu /y)+y/\mu -1),&{\text{for }}p=2\\2\left({\frac {\max(y,0)^{2-p}}{(1-p)(2-p)}}-{\frac {y\mu ^{1-p}}{1-p}}+{\frac {\mu ^{2-p}}{2-p}}\right),&{\text{else}}\end{cases}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="The_Tweedie_cumulant_generating_functions">The Tweedie cumulant generating functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=8" title="Edit section: The Tweedie cumulant generating functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The properties of exponential dispersion models give us two <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>.<sup id="cite_ref-Jørgensen-1997_2-2" class="reference"><a href="#cite_note-Jørgensen-1997-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The first relates the mean value mapping and the variance function to each other, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \tau ^{-1}(\mu )}{\partial \mu }}={\frac {1}{V(\mu )}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <msup> <mi>τ<!-- τ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mi>μ<!-- μ --></mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \tau ^{-1}(\mu )}{\partial \mu }}={\frac {1}{V(\mu )}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e85dabd861bb75fc71cd9ca052ddaf03fbf673d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.535ex; height:6.676ex;" alt="{\displaystyle {\frac {\partial \tau ^{-1}(\mu )}{\partial \mu }}={\frac {1}{V(\mu )}}.}"></span> </p><p>The second shows how the mean value mapping is related to the <a href="/wiki/Cumulant" title="Cumulant">cumulant function</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \kappa (\theta )}{\partial \theta }}=\tau (\theta ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mi>κ<!-- κ --></mi> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mi>θ<!-- θ --></mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>τ<!-- τ --></mi> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \kappa (\theta )}{\partial \theta }}=\tau (\theta ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398be39fd81697e0a38ddd1181d93e196c71b20e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.24ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \kappa (\theta )}{\partial \theta }}=\tau (\theta ).}"></span> </p><p>These equations can be solved to obtain the cumulant function for different cases of the Tweedie models. A cumulant generating function (CGF) may then be obtained from the cumulant function. The additive CGF is generally specified by the equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{*}(s)=\log[\operatorname {E} (e^{sZ})]=\lambda [\kappa (\theta +s)-\kappa (\theta )],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">[</mo> <mi mathvariant="normal">E</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>Z</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mi>λ<!-- λ --></mi> <mo stretchy="false">[</mo> <mi>κ<!-- κ --></mi> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo>+</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>κ<!-- κ --></mi> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{*}(s)=\log[\operatorname {E} (e^{sZ})]=\lambda [\kappa (\theta +s)-\kappa (\theta )],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14c5465bed8147359ff4b22bdb64ab0bbb63f008" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.722ex; height:3.176ex;" alt="{\displaystyle K^{*}(s)=\log[\operatorname {E} (e^{sZ})]=\lambda [\kappa (\theta +s)-\kappa (\theta )],}"></span> and the reproductive CGF by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(s)=\log[\operatorname {E} (e^{sY})]=\lambda [\kappa (\theta +s/\lambda )-\kappa (\theta )],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">[</mo> <mi mathvariant="normal">E</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>Y</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mi>λ<!-- λ --></mi> <mo stretchy="false">[</mo> <mi>κ<!-- κ --></mi> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo>+</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ<!-- λ --></mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>κ<!-- κ --></mi> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(s)=\log[\operatorname {E} (e^{sY})]=\lambda [\kappa (\theta +s/\lambda )-\kappa (\theta )],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95f7fa0639dd1577782b81f5d1097e585274a5f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.223ex; height:3.176ex;" alt="{\displaystyle K(s)=\log[\operatorname {E} (e^{sY})]=\lambda [\kappa (\theta +s/\lambda )-\kappa (\theta )],}"></span> where <i>s</i> is the generating function variable. </p><p>For the additive Tweedie models the CGFs take the form, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{p}^{*}(s;\theta ,\lambda )={\begin{cases}\lambda \kappa _{p}(\theta )[(1+s/\theta )^{\alpha }-1]&\quad p\neq 1,2,\\-\lambda \log(1+s/\theta )&\quad p=2,\\\lambda e^{\theta }(e^{s}-1)&\quad p=1,\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>s</mi> <mo>;</mo> <mi>θ<!-- θ --></mi> <mo>,</mo> <mi>λ<!-- λ --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>λ<!-- λ --></mi> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>θ<!-- θ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msup> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <mspace width="1em" /> <mi>p</mi> <mo>≠<!-- ≠ --></mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>λ<!-- λ --></mi> <mi>log</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mspace width="1em" /> <mi>p</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>λ<!-- λ --></mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ<!-- θ --></mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mspace width="1em" /> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{p}^{*}(s;\theta ,\lambda )={\begin{cases}\lambda \kappa _{p}(\theta )[(1+s/\theta )^{\alpha }-1]&\quad p\neq 1,2,\\-\lambda \log(1+s/\theta )&\quad p=2,\\\lambda e^{\theta }(e^{s}-1)&\quad p=1,\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc5fb576bfad77d907a2af1d28ce410f3580fa4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:51.785ex; height:8.843ex;" alt="{\displaystyle K_{p}^{*}(s;\theta ,\lambda )={\begin{cases}\lambda \kappa _{p}(\theta )[(1+s/\theta )^{\alpha }-1]&\quad p\neq 1,2,\\-\lambda \log(1+s/\theta )&\quad p=2,\\\lambda e^{\theta }(e^{s}-1)&\quad p=1,\end{cases}}}"></span> and for the reproductive models, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{p}(s;\theta ,\lambda )={\begin{cases}\lambda \kappa _{p}(\theta )\left\{\left[1+s/(\theta \lambda )\right]^{\alpha }-1\right\}&\quad p\neq 1,2,\\[1ex]-\lambda \log[1+s/(\theta \lambda )]&\quad p=2,\\[1ex]\lambda e^{\theta }\left(e^{s/\lambda }-1\right)&\quad p=1.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo>;</mo> <mi>θ<!-- θ --></mi> <mo>,</mo> <mi>λ<!-- λ --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.63em 0.63em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>λ<!-- λ --></mi> <msub> <mi>κ<!-- κ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> <mrow> <mo>{</mo> <mrow> <msup> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mi>λ<!-- λ --></mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msup> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> </mtd> <mtd> <mspace width="1em" /> <mi>p</mi> <mo>≠<!-- ≠ --></mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>λ<!-- λ --></mi> <mi>log</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>+</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mi>λ<!-- λ --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> <mtd> <mspace width="1em" /> <mi>p</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>λ<!-- λ --></mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ<!-- θ --></mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ<!-- λ --></mi> </mrow> </msup> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mspace width="1em" /> <mi>p</mi> <mo>=</mo> <mn>1.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{p}(s;\theta ,\lambda )={\begin{cases}\lambda \kappa _{p}(\theta )\left\{\left[1+s/(\theta \lambda )\right]^{\alpha }-1\right\}&\quad p\neq 1,2,\\[1ex]-\lambda \log[1+s/(\theta \lambda )]&\quad p=2,\\[1ex]\lambda e^{\theta }\left(e^{s/\lambda }-1\right)&\quad p=1.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7956dae8bdd3d80e91c4c86ee16f1fbde57cc08" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:55.737ex; height:11.176ex;" alt="{\displaystyle K_{p}(s;\theta ,\lambda )={\begin{cases}\lambda \kappa _{p}(\theta )\left\{\left[1+s/(\theta \lambda )\right]^{\alpha }-1\right\}&\quad p\neq 1,2,\\[1ex]-\lambda \log[1+s/(\theta \lambda )]&\quad p=2,\\[1ex]\lambda e^{\theta }\left(e^{s/\lambda }-1\right)&\quad p=1.\end{cases}}}"></span> </p><p>The additive and reproductive Tweedie models are conventionally denoted by the symbols <i>Tw</i><sup>*</sup><sub><i>p</i></sub>(<i>θ</i>,<i>λ</i>) and <i>Tw</i><sub><i>p</i></sub>(<i>θ</i>,<i>σ</i><sup>2</sup>), respectively. </p><p>The first and second derivatives of the CGFs, with <i>s</i> = 0, yields the mean and variance, respectively. One can thus confirm that for the additive models the variance relates to the mean by the power law, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {var} (Z)\propto \mathrm {E} (Z)^{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>∝<!-- ∝ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo stretchy="false">(</mo> <mi>Z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {var} (Z)\propto \mathrm {E} (Z)^{p}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a102885b0abcbf743e1d12d94d1f16a9628b4f9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.668ex; height:2.843ex;" alt="{\displaystyle \mathrm {var} (Z)\propto \mathrm {E} (Z)^{p}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="The_Tweedie_convergence_theorem">The Tweedie convergence theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=9" title="Edit section: The Tweedie convergence theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Tweedie exponential dispersion models are fundamental in statistical theory consequent to their roles as foci of <a href="/wiki/Convergence_in_distribution" class="mw-redirect" title="Convergence in distribution">convergence</a> for a wide range of statistical processes. Jørgensen <i>et al</i> proved a theorem that specifies the asymptotic behaviour of variance functions known as the <b>Tweedie convergence theorem</b>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This theorem, in technical terms, is stated thus:<sup id="cite_ref-Jørgensen-1997_2-3" class="reference"><a href="#cite_note-Jørgensen-1997-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The unit variance function is regular of order <i>p</i> at zero (or infinity) provided that <span class="texhtml"><i>V</i>(<i>μ</i>) ~ <i>c</i><sub>0</sub><i>μ</i><sup><i>p</i></sup></span> for <i>μ</i> as it approaches zero (or infinity) for all real values of <i>p</i> and <i>c</i><sub>0</sub> > 0. Then for a unit variance function regular of order <i>p</i> at either zero or infinity and for <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\notin (0,1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∉<!-- ∉ --></mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\notin (0,1),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad22fdea8557846f4808895ccac32773ae0368b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.914ex; height:2.843ex;" alt="{\displaystyle p\notin (0,1),}"></span> for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ<!-- μ --></mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67319256f71b2ecddcb2a1f2a58bef0494135e62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mu >0}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39e32ac6c3a581a1db7eab0ffedb5c3a75b545d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.646ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}>0}"></span> we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{-1}\operatorname {ED} (c\mu ,\sigma ^{2}c^{2-p})\rightarrow Tw_{p}(\mu ,c_{0}\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mi>ED</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>c</mi> <mi>μ<!-- μ --></mi> <mo>,</mo> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−<!-- − --></mo> <mi>p</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi>T</mi> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>μ<!-- μ --></mi> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{-1}\operatorname {ED} (c\mu ,\sigma ^{2}c^{2-p})\rightarrow Tw_{p}(\mu ,c_{0}\sigma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dff9eebbe1c288c8e2bbfaa059f9541246a844e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.553ex; height:3.343ex;" alt="{\displaystyle c^{-1}\operatorname {ED} (c\mu ,\sigma ^{2}c^{2-p})\rightarrow Tw_{p}(\mu ,c_{0}\sigma ^{2})}"></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\downarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">↓<!-- ↓ --></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\downarrow 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4cc35cee4d4e84d9f3eeba8d6cf164fc126a69f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.622ex; height:2.509ex;" alt="{\displaystyle c\downarrow 0}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b4b3d854e10af9e53929e1624a7e527ff49496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.945ex; height:1.843ex;" alt="{\displaystyle c\rightarrow \infty }"></span>, respectively, where the convergence is through values of <i>c</i> such that <i>cμ</i> is in the domain of <i>θ</i> and <i>c</i><sup><i>p</i>−2</sup>/<i>σ</i><sup>2</sup> is in the domain of <i>λ</i>. The model must be infinitely divisible as <i>c</i><sup>2−<i>p</i></sup> approaches infinity.<sup id="cite_ref-Jørgensen-1997_2-4" class="reference"><a href="#cite_note-Jørgensen-1997-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In nontechnical terms this theorem implies that any exponential dispersion model that asymptotically manifests a variance-to-mean power law is required to have a variance function that comes within the <a href="/wiki/Attractor" title="Attractor">domain of attraction</a> of a Tweedie model. Almost all distribution functions with finite cumulant generating functions qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form. Hence many probability distributions have variance functions that express this asymptotic behaviour, and the Tweedie distributions become foci of convergence for a wide range of data types.<sup id="cite_ref-Kendal2011b_8-0" class="reference"><a href="#cite_note-Kendal2011b-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Related_distributions">Related distributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=10" title="Edit section: Related distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Tweedie distributions include a number of familiar distributions as well as some unusual ones, each being specified by the <a href="/wiki/Domain_(mathematical_analysis)" title="Domain (mathematical analysis)">domain</a> of the index parameter. We have the </p> <ul><li>extreme stable distribution, <i>p</i> < 0,</li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>, <i>p</i> = 0,</li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a>, <i>p</i> = 1,</li> <li><a href="/wiki/Compound_Poisson_distribution" title="Compound Poisson distribution">compound Poisson–gamma distribution</a>, 1 < <i>p</i> < 2,</li> <li><a href="/wiki/Gamma_distribution" title="Gamma distribution">gamma distribution</a>, <i>p</i> = 2,</li> <li>positive <a href="/wiki/Stable_distribution" title="Stable distribution">stable distributions</a>, 2 < <i>p</i> < 3,</li> <li><a href="/wiki/Inverse_Gaussian_distribution" title="Inverse Gaussian distribution">Inverse Gaussian distribution</a>, <i>p</i> = 3,</li> <li>positive stable distributions, <i>p</i> > 3, and</li> <li>extreme stable distributions, <i>p</i> = <span class="texhtml">∞</span>.</li></ul> <p>For 0 < <i>p</i> < 1 no Tweedie model exists. Note that all <i>stable</i> distributions mean actually <i>generated by stable distributions</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Occurrence_and_applications">Occurrence and applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=11" title="Edit section: Occurrence and applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="The_Tweedie_models_and_Taylor’s_power_law"><span id="The_Tweedie_models_and_Taylor.E2.80.99s_power_law"></span>The Tweedie models and Taylor’s power law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=12" title="Edit section: The Tweedie models and Taylor’s power law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Taylor%27s_law" title="Taylor's law">Taylor's law</a> is an empirical law in <a href="/wiki/Ecology" title="Ecology">ecology</a> that relates the variance of the number of individuals of a species per unit area of habitat to the corresponding mean by a <a href="/wiki/Power-law" class="mw-redirect" title="Power-law">power-law</a> relationship.<sup id="cite_ref-Taylor1961_9-0" class="reference"><a href="#cite_note-Taylor1961-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> For the population count <i>Y</i> with mean <i>μ</i> and variance var(<i>Y</i>), Taylor's law is written, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {var} (Y)=a\mu ^{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>var</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <msup> <mi>μ<!-- μ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {var} (Y)=a\mu ^{p},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b2895cc96f46d2d7e0d67c0fd837d8f81eede6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.32ex; height:2.843ex;" alt="{\displaystyle \operatorname {var} (Y)=a\mu ^{p},}"></span> where <i>a</i> and <i>p</i> are both positive constants. Since L. R. Taylor described this law in 1961 there have been many different explanations offered to explain it, ranging from animal behavior,<sup id="cite_ref-Taylor1961_9-1" class="reference"><a href="#cite_note-Taylor1961-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> a <a href="/wiki/Random_walk" title="Random walk">random walk</a> model,<sup id="cite_ref-Hanski1980_10-0" class="reference"><a href="#cite_note-Hanski1980-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> a <a href="/wiki/Birth%E2%80%93death_process" title="Birth–death process">stochastic birth, death, immigration and emigration model</a>,<sup id="cite_ref-Anderson1961_11-0" class="reference"><a href="#cite_note-Anderson1961-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> to a consequence of equilibrium and non-equilibrium <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>.<sup id="cite_ref-Fronczak2010_12-0" class="reference"><a href="#cite_note-Fronczak2010-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> No consensus exists as to an explanation for this model. </p><p>Since Taylor's law is mathematically identical to the variance-to-mean power law that characterizes the Tweedie models, it seemed reasonable to use these models and the Tweedie convergence theorem to explain the observed clustering of animals and plants associated with Taylor's law.<sup id="cite_ref-Kendal2002_13-0" class="reference"><a href="#cite_note-Kendal2002-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kendal2004_14-0" class="reference"><a href="#cite_note-Kendal2004-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> The majority of the observed values for the power-law exponent <i>p</i> have fallen in the interval (1,2) and so the Tweedie compound Poisson–gamma distribution would seem applicable. Comparison of the <a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">empirical distribution function</a> to the theoretical compound Poisson–gamma distribution has provided a means to verify consistency of this hypothesis.<sup id="cite_ref-Kendal2002_13-1" class="reference"><a href="#cite_note-Kendal2002-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Whereas conventional models for Taylor's law have tended to involve <i><a href="/wiki/Ad_hoc" title="Ad hoc">ad hoc</a></i> animal behavioral or <a href="/wiki/Population_dynamics" title="Population dynamics">population dynamic</a> assumptions, the Tweedie convergence theorem would imply that Taylor's law results from a general mathematical convergence effect much as how the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a> governs the convergence behavior of certain types of random data. Indeed, any mathematical model, approximation or simulation that is designed to yield Taylor's law (on the basis of this theorem) is required to converge to the form of the Tweedie models.<sup id="cite_ref-Kendal2011b_8-1" class="reference"><a href="#cite_note-Kendal2011b-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Tweedie_convergence_and_1/f_noise"><span id="Tweedie_convergence_and_1.2Ff_noise"></span>Tweedie convergence and 1/<i>f</i> noise</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=13" title="Edit section: Tweedie convergence and 1/f noise"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Pink_noise" title="Pink noise">Pink noise</a>, or 1/<i>f</i> noise, refers to a pattern of noise characterized by a power-law relationship between its intensities <i>S</i>(<i>f</i>) at different frequencies <i>f</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(f)\propto {\frac {1}{f^{\gamma }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>∝<!-- ∝ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ<!-- γ --></mi> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(f)\propto {\frac {1}{f^{\gamma }}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d3b70066e0470ff44dedb068edea55890d75f09" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.614ex; height:5.676ex;" alt="{\displaystyle S(f)\propto {\frac {1}{f^{\gamma }}},}"></span> where the dimensionless exponent <i>γ</i> ∈ [0,1]. It is found within a diverse number of natural processes.<sup id="cite_ref-Dutta1981_15-0" class="reference"><a href="#cite_note-Dutta1981-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Many different explanations for 1/<i>f</i> noise exist, a widely held hypothesis is based on <a href="/wiki/Self-organized_criticality" title="Self-organized criticality">Self-organized criticality</a> where dynamical systems close to a <a href="/wiki/Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">critical point</a> are thought to manifest <a href="/wiki/Scale-invariance" class="mw-redirect" title="Scale-invariance">scale-invariant</a> spatial and/or temporal behavior. </p><p>In this subsection a mathematical connection between 1/<i>f</i> noise and the Tweedie variance-to-mean power law will be described. To begin, we first need to introduce <a href="/wiki/Self-similar_process" title="Self-similar process">self-similar processes</a>: For the sequence of numbers <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=(Y_{i}:i=0,1,2,\ldots ,N)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…<!-- … --></mo> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=(Y_{i}:i=0,1,2,\ldots ,N)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f472b1c206affdc8a5d2aa8e03978b23dbd53cb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.466ex; height:2.843ex;" alt="{\displaystyle Y=(Y_{i}:i=0,1,2,\ldots ,N)}"></span> with mean <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mu }}=\operatorname {E} (Y_{i}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ<!-- μ --></mi> <mo>^<!-- ^ --></mo> </mover> </mrow> </mrow> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mu }}=\operatorname {E} (Y_{i}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50f2dcd80e6cefe001c76727abb66040eb7bdf5e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.699ex; height:2.843ex;" alt="{\displaystyle {\widehat {\mu }}=\operatorname {E} (Y_{i}),}"></span> deviations <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}=Y_{i}-{\widehat {\mu }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ<!-- μ --></mi> <mo>^<!-- ^ --></mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}=Y_{i}-{\widehat {\mu }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2a5e90f8c5e906cdfb74366020de93f435e9e4f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.087ex; height:2.843ex;" alt="{\displaystyle y_{i}=Y_{i}-{\widehat {\mu }},}"></span> variance <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\sigma }}^{2}=\operatorname {E} (y_{i}^{2}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ<!-- σ --></mi> <mo>^<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\sigma }}^{2}=\operatorname {E} (y_{i}^{2}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e837c8b7a574305ef1d4105b5d70eb22719a40" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.736ex; height:3.509ex;" alt="{\displaystyle {\widehat {\sigma }}^{2}=\operatorname {E} (y_{i}^{2}),}"></span> and autocorrelation function <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(k)={\frac {\operatorname {E} (y_{i},y_{i+k})}{\operatorname {E} (y_{i}^{2})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(k)={\frac {\operatorname {E} (y_{i},y_{i+k})}{\operatorname {E} (y_{i}^{2})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b15e11df4ff7dedf77403f685a339176cbb6edce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.443ex; height:6.843ex;" alt="{\displaystyle r(k)={\frac {\operatorname {E} (y_{i},y_{i+k})}{\operatorname {E} (y_{i}^{2})}}}"></span> with lag <i>k</i>, if the <a href="/wiki/Autocorrelation" title="Autocorrelation">autocorrelation</a> of this sequence has the long range behavior <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(k)\sim k^{-d}L(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>∼<!-- ∼ --></mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mi>d</mi> </mrow> </msup> <mi>L</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(k)\sim k^{-d}L(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2fea8c1818817408527905196df8b32f06a3075" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.353ex; height:3.176ex;" alt="{\displaystyle r(k)\sim k^{-d}L(k)}"></span> as <i>k</i><span class="texhtml">→∞</span> and where <i>L</i>(<i>k</i>) is a slowly varying function at large values of <i>k</i>, this sequence is called a self-similar process.<sup id="cite_ref-Leland1994_16-0" class="reference"><a href="#cite_note-Leland1994-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>The <b>method of expanding bins</b> can be used to analyze self-similar processes. Consider a set of equal-sized non-overlapping bins that divides the original sequence of <i>N</i> elements into groups of <i>m</i> equal-sized segments (<i>N/m</i> is integer) so that new reproductive sequences, based on the mean values, can be defined: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}^{(m)}=\left(Y_{im-m+1}+\cdots +Y_{im}\right)/m.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> <mo>−<!-- − --></mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{i}^{(m)}=\left(Y_{im-m+1}+\cdots +Y_{im}\right)/m.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55bf89493afa3586312489230e74f02ac674b7f1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.411ex; height:3.676ex;" alt="{\displaystyle Y_{i}^{(m)}=\left(Y_{im-m+1}+\cdots +Y_{im}\right)/m.}"></span> </p><p>The variance determined from this sequence will scale as the bin size changes such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {var} [Y^{(m)}]={\widehat {\sigma }}^{2}m^{-d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>var</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">[</mo> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ<!-- σ --></mi> <mo>^<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {var} [Y^{(m)}]={\widehat {\sigma }}^{2}m^{-d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97ab41f8e335abb34bf2e4b1a3bf9b5c755880fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.343ex; height:3.343ex;" alt="{\displaystyle \operatorname {var} [Y^{(m)}]={\widehat {\sigma }}^{2}m^{-d}}"></span> if and only if the autocorrelation has the limiting form<sup id="cite_ref-Tsybakov1997_17-0" class="reference"><a href="#cite_note-Tsybakov1997-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{k\to \infty }r(k)/k^{-d}=(2-d)(1-d)/2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munder> <mi>r</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mi>d</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−<!-- − --></mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>d</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{k\to \infty }r(k)/k^{-d}=(2-d)(1-d)/2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beae55e69507b7605418debb6ba7242e951da057" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.47ex; height:4.343ex;" alt="{\displaystyle \lim _{k\to \infty }r(k)/k^{-d}=(2-d)(1-d)/2.}"></span> </p><p>One can also construct a set of corresponding additive sequences <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{i}^{(m)}=mY_{i}^{(m)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mi>m</mi> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{i}^{(m)}=mY_{i}^{(m)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c69e06b5d1f0022a8106de5de1514bac455f878" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.303ex; height:3.676ex;" alt="{\displaystyle Z_{i}^{(m)}=mY_{i}^{(m)},}"></span> based on the expanding bins, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{i}^{(m)}=(Y_{im-m+1}+\cdots +Y_{im}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> <mo>−<!-- − --></mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯<!-- ⋯ --></mo> <mo>+</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{i}^{(m)}=(Y_{im-m+1}+\cdots +Y_{im}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e23a2ee54f5cce58e10900ac79d095f6cd417da1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.629ex; height:3.676ex;" alt="{\displaystyle Z_{i}^{(m)}=(Y_{im-m+1}+\cdots +Y_{im}).}"></span> </p><p>Provided the autocorrelation function exhibits the same behavior, the additive sequences will obey the relationship <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {var} [Z_{i}^{(m)}]=m^{2}\operatorname {var} [Y^{(m)}]=\left({\frac {{\widehat {\sigma }}^{2}}{{\widehat {\mu }}^{2-d}}}\right)\operatorname {E} [Z_{i}^{(m)}]^{2-d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>var</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">[</mo> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>var</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">[</mo> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ<!-- σ --></mi> <mo>^<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ<!-- μ --></mi> <mo>^<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−<!-- − --></mo> <mi>d</mi> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">E</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">[</mo> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−<!-- − --></mo> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {var} [Z_{i}^{(m)}]=m^{2}\operatorname {var} [Y^{(m)}]=\left({\frac {{\widehat {\sigma }}^{2}}{{\widehat {\mu }}^{2-d}}}\right)\operatorname {E} [Z_{i}^{(m)}]^{2-d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31c693170f5697765067347d370a3bf179fa55c0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.627ex; height:7.509ex;" alt="{\displaystyle \operatorname {var} [Z_{i}^{(m)}]=m^{2}\operatorname {var} [Y^{(m)}]=\left({\frac {{\widehat {\sigma }}^{2}}{{\widehat {\mu }}^{2-d}}}\right)\operatorname {E} [Z_{i}^{(m)}]^{2-d}}"></span> </p><p>Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ<!-- μ --></mi> <mo>^<!-- ^ --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0771ade78a4446e1f7bf6900e637b788c0cbad92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.019ex; width:1.43ex; height:2.843ex;" alt="{\displaystyle {\widehat {\mu }}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\sigma }}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ<!-- σ --></mi> <mo>^<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\sigma }}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5afe69bbdc03070ef019f43fe451b847d2acf1ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.843ex;" alt="{\displaystyle {\widehat {\sigma }}^{2}}"></span> are constants this relationship constitutes a variance-to-mean power law, with <i>p</i> = 2 - <i>d</i>.<sup id="cite_ref-Kendal2011b_8-2" class="reference"><a href="#cite_note-Kendal2011b-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kendal2007_18-0" class="reference"><a href="#cite_note-Kendal2007-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Logical_biconditional" title="Logical biconditional">biconditional</a> relationship above between the variance-to-mean power law and power law autocorrelation function, and the <a href="/wiki/Wiener%E2%80%93Khinchin_theorem" title="Wiener–Khinchin theorem">Wiener–Khinchin theorem</a><sup id="cite_ref-McQuarrie1976_19-0" class="reference"><a href="#cite_note-McQuarrie1976-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> imply that any sequence that exhibits a variance-to-mean power law by the method of expanding bins will also manifest 1/<i>f</i> noise, and vice versa. Moreover, the Tweedie convergence theorem, by virtue of its central limit-like effect of generating distributions that manifest variance-to-mean power functions, will also generate processes that manifest 1/<i>f</i> noise.<sup id="cite_ref-Kendal2011b_8-3" class="reference"><a href="#cite_note-Kendal2011b-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The Tweedie convergence theorem thus provides an alternative explanation for the origin of 1/<i>f</i> noise, based its central limit-like effect. </p><p>Much as the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a> requires certain kinds of random processes to have as a focus of their convergence the <a href="/wiki/Normal_distribution" title="Normal distribution">Gaussian distribution</a> and thus express <a href="/wiki/White_noise" title="White noise">white noise</a>, the Tweedie convergence theorem requires certain non-Gaussian processes to have as a focus of convergence the Tweedie distributions that express 1/<i>f</i> noise.<sup id="cite_ref-Kendal2011b_8-4" class="reference"><a href="#cite_note-Kendal2011b-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="The_Tweedie_models_and_multifractality">The Tweedie models and multifractality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=14" title="Edit section: The Tweedie models and multifractality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>From the properties of self-similar processes, the power-law exponent <i>p</i> = 2 - <i>d</i> is related to the <a href="/wiki/Hurst_exponent" title="Hurst exponent">Hurst exponent</a> <i>H</i> and the <a href="/wiki/Fractal_dimension" title="Fractal dimension">fractal dimension</a> <i>D</i> by<sup id="cite_ref-Tsybakov1997_17-1" class="reference"><a href="#cite_note-Tsybakov1997-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=2-H=2-p/2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mn>2</mn> <mo>−<!-- − --></mo> <mi>H</mi> <mo>=</mo> <mn>2</mn> <mo>−<!-- − --></mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=2-H=2-p/2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c3fbf8d9ff8f3f7f15e0dba388c065553db5ac0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.332ex; height:2.843ex;" alt="{\displaystyle D=2-H=2-p/2.}"></span> </p><p>A one-dimensional data sequence of self-similar data may demonstrate a variance-to-mean power law with local variations in the value of <i>p</i> and hence in the value of <i>D</i>. When fractal structures manifest local variations in fractal dimension, they are said to be <a href="/wiki/Multifractal_system" title="Multifractal system">multifractals</a>. Examples of data sequences that exhibit local variations in <i>p</i> like this include the eigenvalue deviations of the <a href="/wiki/Random_matrix" title="Random matrix">Gaussian Orthogonal and Unitary Ensembles</a>.<sup id="cite_ref-Kendal2011b_8-5" class="reference"><a href="#cite_note-Kendal2011b-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The Tweedie compound Poisson–gamma distribution has served to model multifractality based on local variations in the Tweedie exponent <i>α</i>. Consequently, in conjunction with the variation of <i>α</i>, the Tweedie convergence theorem can be viewed as having a role in the genesis of such multifractals. </p><p>The variation of <i>α</i> has been found to obey the asymmetric <a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace distribution</a> in certain cases.<sup id="cite_ref-Kendal2014_20-0" class="reference"><a href="#cite_note-Kendal2014-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> This distribution has been shown to be a member of the family of geometric Tweedie models,<sup id="cite_ref-Jørgensen2011_21-0" class="reference"><a href="#cite_note-Jørgensen2011-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> that manifest as limiting distributions in a convergence theorem for geometric dispersion models. </p> <div class="mw-heading mw-heading3"><h3 id="Regional_organ_blood_flow">Regional organ blood flow</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=15" title="Edit section: Regional organ blood flow"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Regional organ blood flow has been traditionally assessed by the injection of <a href="/wiki/Isotopic_labeling" title="Isotopic labeling">radiolabelled</a> <a href="/wiki/Polyethylene_microspheres" class="mw-redirect" title="Polyethylene microspheres">polyethylene microspheres</a> into the arterial circulation of animals, of a size that they become entrapped within the <a href="/wiki/Microcirculation" title="Microcirculation">microcirculation</a> of organs. The organ to be assessed is then divided into equal-sized cubes and the amount of radiolabel within each cube is evaluated by <a href="/wiki/Liquid_scintillation_counting" title="Liquid scintillation counting">liquid scintillation counting</a> and recorded. The amount of radioactivity within each cube is taken to reflect the blood flow through that sample at the time of injection. It is possible to evaluate adjacent cubes from an organ in order to additively determine the blood flow through larger regions. Through the work of <b>J B Bassingthwaighte</b> and others an empirical power law has been derived between the relative dispersion of blood flow of tissue samples (<i>RD</i> = standard deviation/mean) of mass <i>m</i> relative to reference-sized samples:<sup id="cite_ref-Bassingthwaighte1989_22-0" class="reference"><a href="#cite_note-Bassingthwaighte1989-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle RD(m)=RD(m_{\text{ref}})\left({\frac {m}{m_{\text{ref}}}}\right)^{1-D_{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>D</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>R</mi> <mi>D</mi> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>ref</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>ref</mtext> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−<!-- − --></mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RD(m)=RD(m_{\text{ref}})\left({\frac {m}{m_{\text{ref}}}}\right)^{1-D_{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/162f042640b249a2c1be1e9fb3d86685398619d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.176ex; height:6.676ex;" alt="{\displaystyle RD(m)=RD(m_{\text{ref}})\left({\frac {m}{m_{\text{ref}}}}\right)^{1-D_{s}}}"></span> </p><p>This power law exponent <i>D<sub>s</sub></i> has been called a fractal dimension. <b>Bassingthwaighte's power law</b> can be shown to directly relate to the variance-to-mean power law. Regional organ blood flow can thus be modelled by the Tweedie compound Poisson–gamma distribution.,<sup id="cite_ref-Kendal2001_23-0" class="reference"><a href="#cite_note-Kendal2001-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> In this model tissue sample could be considered to contain a random (Poisson) distributed number of entrapment sites, each with <a href="/wiki/Gamma_distribution" title="Gamma distribution">gamma distributed</a> blood flow. Blood flow at this microcirculatory level has been observed to obey a gamma distribution,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> thus providing support for this hypothesis. </p> <div class="mw-heading mw-heading3"><h3 id="Cancer_metastasis">Cancer metastasis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=16" title="Edit section: Cancer metastasis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The "experimental cancer <a href="/wiki/Metastasis" title="Metastasis">metastasis</a> assay"<sup id="cite_ref-Fidler1977_25-0" class="reference"><a href="#cite_note-Fidler1977-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> has some resemblance to the above method to measure regional blood flow. Groups of <a href="/wiki/Syngeneic" class="mw-redirect" title="Syngeneic">syngeneic</a> and age matched mice are given intravenous injections of equal-sized aliquots of suspensions of cloned cancer cells and then after a set period of time their lungs are removed and the number of cancer metastases enumerated within each pair of lungs. If other groups of mice are injected with different cancer cell <a href="/wiki/Clone_(cell_biology)" title="Clone (cell biology)">clones</a> then the number of metastases per group will differ in accordance with the metastatic potentials of the clones. It has been long recognized that there can be considerable intraclonal variation in the numbers of metastases per mouse despite the best attempts to keep the experimental conditions within each clonal group uniform.<sup id="cite_ref-Fidler1977_25-1" class="reference"><a href="#cite_note-Fidler1977-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> This variation is larger than would be expected on the basis of a <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a> of numbers of metastases per mouse in each clone and when the variance of the number of metastases per mouse was plotted against the corresponding mean a power law was found.<sup id="cite_ref-Kendal1987_26-0" class="reference"><a href="#cite_note-Kendal1987-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>The variance-to-mean power law for metastases was found to also hold for <b>spontaneous murine metastases</b><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> and for cases series of human metastases.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Since hematogenous metastasis occurs in direct relationship to regional blood flow<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> and videomicroscopic studies indicate that the passage and entrapment of cancer cells within the circulation appears analogous to the microsphere experiments<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> it seemed plausible to propose that the variation in numbers of hematogenous metastases could reflect heterogeneity in regional organ blood flow.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> The blood flow model was based on the Tweedie compound Poisson–gamma distribution, a distribution governing a continuous random variable. For that reason in the metastasis model it was assumed that blood flow was governed by that distribution and that the number of regional metastases occurred as a <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a> for which the intensity was directly proportional to blood flow. This led to the description of the Poisson negative binomial (PNB) distribution as a <a href="/wiki/Discrete_probability_distribution" class="mw-redirect" title="Discrete probability distribution">discrete equivalent</a> to the Tweedie compound Poisson–gamma distribution. The <a href="/wiki/Probability-generating_function" title="Probability-generating function">probability generating function</a> for the PNB distribution is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(s)=\exp \left[\lambda {\frac {\alpha -1}{\alpha }}\left({\frac {\theta }{\alpha -1}}\right)^{\alpha }\left\{\left(1-{\frac {1}{\theta }}+{\frac {s}{\theta }}\right)^{\alpha }-1\right\}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡<!-- --></mo> <mrow> <mo>[</mo> <mrow> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>α<!-- α --></mi> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ<!-- θ --></mi> <mrow> <mi>α<!-- α --></mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msup> <mrow> <mo>{</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>θ<!-- θ --></mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>θ<!-- θ --></mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msup> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(s)=\exp \left[\lambda {\frac {\alpha -1}{\alpha }}\left({\frac {\theta }{\alpha -1}}\right)^{\alpha }\left\{\left(1-{\frac {1}{\theta }}+{\frac {s}{\theta }}\right)^{\alpha }-1\right\}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00a7797eee272058f54459094a8e27e812860f40" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.896ex; height:6.176ex;" alt="{\displaystyle G(s)=\exp \left[\lambda {\frac {\alpha -1}{\alpha }}\left({\frac {\theta }{\alpha -1}}\right)^{\alpha }\left\{\left(1-{\frac {1}{\theta }}+{\frac {s}{\theta }}\right)^{\alpha }-1\right\}\right]}"></span> </p><p>The relationship between the mean and variance of the PNB distribution is then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {var} (Y)=a\operatorname {E} (Y)^{b}+\operatorname {E} (Y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>var</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi mathvariant="normal">E</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>Y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">E</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {var} (Y)=a\operatorname {E} (Y)^{b}+\operatorname {E} (Y),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/895d0afc6c2cf0fd08854e7ba28aa3ed993b03c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.355ex; height:3.176ex;" alt="{\displaystyle \operatorname {var} (Y)=a\operatorname {E} (Y)^{b}+\operatorname {E} (Y),}"></span> which, in the range of many experimental metastasis assays, would be indistinguishable from the variance-to-mean power law. For sparse data, however, this discrete variance-to-mean relationship would behave more like that of a Poisson distribution where the variance equaled the mean. </p> <div class="mw-heading mw-heading3"><h3 id="Genomic_structure_and_evolution">Genomic structure and evolution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=17" title="Edit section: Genomic structure and evolution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The local density of <a href="/wiki/Single-nucleotide_polymorphism" title="Single-nucleotide polymorphism">Single Nucleotide Polymorphisms</a> (SNPs) within the <a href="/wiki/Human_genome" title="Human genome">human genome</a>, as well as that of <a href="/wiki/Gene" title="Gene">genes</a>, appears to cluster in accord with the variance-to-mean power law and the Tweedie compound Poisson–gamma distribution.<sup id="cite_ref-Kendal2003_32-0" class="reference"><a href="#cite_note-Kendal2003-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-KendalGenes_33-0" class="reference"><a href="#cite_note-KendalGenes-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> In the case of SNPs their observed density reflects the assessment techniques, the availability of genomic sequences for analysis, and the <a href="/wiki/Nucleotide_diversity" title="Nucleotide diversity">nucleotide heterozygosity</a>.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> The first two factors reflect ascertainment errors inherent to the collection methods, the latter factor reflects an intrinsic property of the genome. </p><p>In the <a href="/wiki/Coalescent_theory" title="Coalescent theory">coalescent model</a> of population genetics each genetic locus has its own unique history. Within the evolution of a population from some species some genetic loci could presumably be traced back to a relatively <a href="/wiki/Most_recent_common_ancestor" title="Most recent common ancestor">recent common ancestor</a> whereas other loci might have more ancient <a href="/wiki/Genetic_genealogy" title="Genetic genealogy">genealogies</a>. More ancient genomic segments would have had more time to accumulate SNPs and to experience <a href="/wiki/Genetic_recombination" title="Genetic recombination">recombination</a>. <b>R R Hudson</b> has proposed a model where recombination could cause variation in the time to <a href="/wiki/Most_recent_common_ancestor" title="Most recent common ancestor">most common recent ancestor</a> for different genomic segments.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> A high recombination rate could cause a chromosome to contain a large number of small segments with less correlated genealogies. </p><p>Assuming a constant background rate of mutation the number of SNPs per genomic segment would accumulate proportionately to the time to the most recent common ancestor. Current <a href="/wiki/Population_genetics" title="Population genetics">population genetic theory</a> would indicate that these times would be <a href="/wiki/Gamma_distribution" title="Gamma distribution">gamma distributed</a>, on average.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> The Tweedie compound Poisson–gamma distribution would suggest a model whereby the SNP map would consist of multiple small genomic segments with the mean number of SNPs per segment would be gamma distributed as per Hudson's model. </p><p>The distribution of genes within the human genome also demonstrated a variance-to-mean power law, when the method of expanding bins was used to determine the corresponding variances and means.<sup id="cite_ref-KendalGenes_33-1" class="reference"><a href="#cite_note-KendalGenes-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Similarly the number of genes per enumerative bin was found to obey a Tweedie compound Poisson–gamma distribution. This probability distribution was deemed compatible with two different biological models: the <b>microarrangement model</b> where the number of genes per unit genomic length was determined by the sum of a random number of smaller genomic segments derived by random breakage and reconstruction of protochormosomes. These smaller segments would be assumed to carry on average a gamma distributed number of genes. </p><p>In the alternative <b>gene cluster model</b>, genes would be distributed randomly within the protochromosomes. Over large evolutionary timescales there would occur <a href="/wiki/Gene_duplication" title="Gene duplication">tandem duplication</a>, <a href="/wiki/Mutation" title="Mutation">mutations, insertions, deletions</a> and <a href="/wiki/Chromosomal_rearrangement" title="Chromosomal rearrangement">rearrangements</a> that could affect the genes through a stochastic <a href="/wiki/Birth%E2%80%93death_process" title="Birth–death process">birth, death and immigration process</a> to yield the Tweedie compound Poisson–gamma distribution. </p><p>Both these mechanisms would implicate <a href="/wiki/Neutral_theory_of_molecular_evolution" title="Neutral theory of molecular evolution">neutral evolutionary processes</a> that would result in regional clustering of genes. </p> <div class="mw-heading mw-heading3"><h3 id="Random_matrix_theory">Random matrix theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=18" title="Edit section: Random matrix theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b><a href="/wiki/Random_matrix" title="Random matrix">Gaussian unitary ensemble</a></b> (GUE) consists of complex <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian matrices</a> that are invariant under <a href="/wiki/Unitary_transformation" title="Unitary transformation">unitary transformations</a> whereas the <b><a href="/wiki/Random_matrix" title="Random matrix">Gaussian orthogonal ensemble</a></b> (GOE) consists of real symmetric matrices invariant under <a href="/wiki/Orthogonal_transformation" title="Orthogonal transformation">orthogonal transformations</a>. The ranked <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a> <i>E<sub>n</sub></i> from these random matrices obey <b><a href="/wiki/Wigner_semicircle_distribution" title="Wigner semicircle distribution">Wigner's semicircular distribution</a></b>: For a <i>N</i>×<i>N</i> matrix the average density for eigenvalues of size <i>E</i> will be <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\rho }}(E)={\begin{cases}{\sqrt {2N-E^{2}}}/\pi &\quad \left\vert E\right\vert <{\sqrt {2N}}\\0&\quad \left\vert E\right\vert >{\sqrt {2N}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ<!-- ρ --></mi> <mo stretchy="false">¯<!-- ¯ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>N</mi> <mo>−<!-- − --></mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>π<!-- π --></mi> </mtd> <mtd> <mspace width="1em" /> <mrow> <mo>|</mo> <mi>E</mi> <mo>|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>N</mi> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mspace width="1em" /> <mrow> <mo>|</mo> <mi>E</mi> <mo>|</mo> </mrow> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>N</mi> </msqrt> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\rho }}(E)={\begin{cases}{\sqrt {2N-E^{2}}}/\pi &\quad \left\vert E\right\vert <{\sqrt {2N}}\\0&\quad \left\vert E\right\vert >{\sqrt {2N}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1393997cdfdc60eea7c231f26f61214ae2630e8b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.261ex; height:6.509ex;" alt="{\displaystyle {\bar {\rho }}(E)={\begin{cases}{\sqrt {2N-E^{2}}}/\pi &\quad \left\vert E\right\vert <{\sqrt {2N}}\\0&\quad \left\vert E\right\vert >{\sqrt {2N}}\end{cases}}}"></span> as <i>E</i><span class="texhtml">→ ∞</span>. Integration of the semicircular rule provides the number of eigenvalues on average less than <i>E</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\eta }}(E)={\frac {1}{2\pi }}\left[E{\sqrt {2N-E^{2}}}+2N\arcsin \left({\frac {E}{\sqrt {2N}}}\right)+\pi N\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>η<!-- η --></mi> <mo stretchy="false">¯<!-- ¯ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π<!-- π --></mi> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>N</mi> <mo>−<!-- − --></mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mn>2</mn> <mi>N</mi> <mi>arcsin</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <msqrt> <mn>2</mn> <mi>N</mi> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>π<!-- π --></mi> <mi>N</mi> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\eta }}(E)={\frac {1}{2\pi }}\left[E{\sqrt {2N-E^{2}}}+2N\arcsin \left({\frac {E}{\sqrt {2N}}}\right)+\pi N\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e30bb44ccfe4a2e7f07d741bf9e8bd751074aa7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:56.279ex; height:6.509ex;" alt="{\displaystyle {\bar {\eta }}(E)={\frac {1}{2\pi }}\left[E{\sqrt {2N-E^{2}}}+2N\arcsin \left({\frac {E}{\sqrt {2N}}}\right)+\pi N\right].}"></span> </p><p>The ranked eigenvalues can be <b>unfolded</b>, or renormalized, with the equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{n}={\bar {\eta }}(E)=\int _{-\infty }^{E_{n}}\,dE'{\bar {\rho }}(E').}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>η<!-- η --></mi> <mo stretchy="false">¯<!-- ¯ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>E</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ<!-- ρ --></mi> <mo stretchy="false">¯<!-- ¯ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{n}={\bar {\eta }}(E)=\int _{-\infty }^{E_{n}}\,dE'{\bar {\rho }}(E').}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/030f38ad4a406d499c005b1de6065f7158a73bf0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.041ex; height:6.343ex;" alt="{\displaystyle e_{n}={\bar {\eta }}(E)=\int _{-\infty }^{E_{n}}\,dE'{\bar {\rho }}(E').}"></span> </p><p>This removes the trend of the sequence from the fluctuating portion. If we look at the absolute value of the difference between the actual and expected cumulative number of eigenvalues <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|{\bar {D}}_{n}\right|=\left|n-{\bar {\eta }}(E_{n})\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo stretchy="false">¯<!-- ¯ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>η<!-- η --></mi> <mo stretchy="false">¯<!-- ¯ --></mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\bar {D}}_{n}\right|=\left|n-{\bar {\eta }}(E_{n})\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae747e2319e23555592f640e72df39e904058c04" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.111ex; height:3.176ex;" alt="{\displaystyle \left|{\bar {D}}_{n}\right|=\left|n-{\bar {\eta }}(E_{n})\right|}"></span> we obtain a sequence of <b>eigenvalue fluctuations</b> which, using the method of expanding bins, reveals a variance-to-mean power law.<sup id="cite_ref-Kendal2011b_8-6" class="reference"><a href="#cite_note-Kendal2011b-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The eigenvalue fluctuations of both the GUE and the GOE manifest this power law with the power law exponents ranging between 1 and 2, and they similarly manifest 1/<i>f</i> noise spectra. These eigenvalue fluctuations also correspond to the Tweedie compound Poisson–gamma distribution and they exhibit multifractality.<sup id="cite_ref-Kendal2011b_8-7" class="reference"><a href="#cite_note-Kendal2011b-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="The_distribution_of_prime_numbers">The distribution of <a href="/wiki/Prime_number" title="Prime number">prime numbers</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=19" title="Edit section: The distribution of prime numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>second <a href="/wiki/Chebyshev_function" title="Chebyshev function">Chebyshev function</a></b> <i>ψ</i>(<i>x</i>) is given by, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)=\sum _{{\widehat {p\,}}^{k}\leq x}\log {\widehat {p\,}}=\sum _{n\leq x}\Lambda (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ<!-- ψ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>p</mi> <mspace width="thinmathspace" /> </mrow> <mo>^<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>≤<!-- ≤ --></mo> <mi>x</mi> </mrow> </munder> <mi>log</mi> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>p</mi> <mspace width="thinmathspace" /> </mrow> <mo>^<!-- ^ --></mo> </mover> </mrow> </mrow> <mo>=</mo> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≤<!-- ≤ --></mo> <mi>x</mi> </mrow> </munder> <mi mathvariant="normal">Λ<!-- Λ --></mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)=\sum _{{\widehat {p\,}}^{k}\leq x}\log {\widehat {p\,}}=\sum _{n\leq x}\Lambda (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a016373edffd4861b3635f1f62e393919a1b97f1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:29.003ex; height:6.676ex;" alt="{\displaystyle \psi (x)=\sum _{{\widehat {p\,}}^{k}\leq x}\log {\widehat {p\,}}=\sum _{n\leq x}\Lambda (n)}"></span> where the summation extends over all prime powers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {p\,}}^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>p</mi> <mspace width="thinmathspace" /> </mrow> <mo>^<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {p\,}}^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27f7f79c8b777adac9618bdc5f09c721136f128c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.735ex; height:3.176ex;" alt="{\displaystyle {\widehat {p\,}}^{k}}"></span> not exceeding <i>x</i>, <i>x</i> runs over the positive real numbers, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ<!-- Λ --></mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/763b9c503bc0ec2109ea1031a176850d169ea833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.817ex; height:2.843ex;" alt="{\displaystyle \Lambda (n)}"></span> is the <a href="/wiki/Von_Mangoldt_function" title="Von Mangoldt function">von Mangoldt function</a>. The function <i>ψ</i>(<i>x</i>) is related to the <a href="/wiki/Prime-counting_function" title="Prime-counting function">prime-counting function</a> <i>π</i>(<i>x</i>), and as such provides information with regards to the distribution of prime numbers amongst the real numbers. It is asymptotic to <i>x</i>, a statement equivalent to the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a> and it can also be shown to be related to the zeros of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> located on the critical strip <i>ρ</i>, where the real part of the zeta zero <i>ρ</i> is between 0 and 1. Then <i>ψ</i> expressed for <i>x</i> greater than one can be written: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\ln 2\pi -{\frac {1}{2}}\ln(1-x^{-2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ<!-- ψ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>−<!-- − --></mo> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ<!-- ρ --></mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ<!-- ρ --></mi> </mrow> </msup> <mi>ρ<!-- ρ --></mi> </mfrac> </mrow> <mo>−<!-- − --></mo> <mi>ln</mi> <mo>⁡<!-- --></mo> <mn>2</mn> <mi>π<!-- π --></mi> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\ln 2\pi -{\frac {1}{2}}\ln(1-x^{-2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568a264ecc9302fab6ad68887af66279361f1740" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:44.266ex; height:6.676ex;" alt="{\displaystyle \psi _{0}(x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\ln 2\pi -{\frac {1}{2}}\ln(1-x^{-2})}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{0}(x)=\lim _{\varepsilon \rightarrow 0}{\frac {\psi (x-\varepsilon )+\psi (x+\varepsilon )}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ<!-- ψ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ε<!-- ε --></mi> <mo stretchy="false">→<!-- → --></mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ψ<!-- ψ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−<!-- − --></mo> <mi>ε<!-- ε --></mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ψ<!-- ψ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>ε<!-- ε --></mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{0}(x)=\lim _{\varepsilon \rightarrow 0}{\frac {\psi (x-\varepsilon )+\psi (x+\varepsilon )}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f728b49936843e866c4c3eddbe33a7aaf7caaa62" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.899ex; height:5.843ex;" alt="{\displaystyle \psi _{0}(x)=\lim _{\varepsilon \rightarrow 0}{\frac {\psi (x-\varepsilon )+\psi (x+\varepsilon )}{2}}.}"></span> </p><p>The <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> states that the <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">nontrivial zeros</a> of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> all have <a href="/wiki/Real_part" class="mw-redirect" title="Real part">real part</a> <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>. These zeta function zeros are related to the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">distribution of prime numbers</a>. <a href="/wiki/Lowell_Schoenfeld" title="Lowell Schoenfeld"><b>Schoenfeld</b></a><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> has shown that if the Riemann hypothesis is true then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (x)=\left\vert \psi (x)-x\right\vert <{\sqrt {x}}\log ^{2}(x)/(8\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>ψ<!-- ψ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <msup> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>8</mn> <mi>π<!-- π --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (x)=\left\vert \psi (x)-x\right\vert <{\sqrt {x}}\log ^{2}(x)/(8\pi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e9ebec30762ebe7353a5ed529027e2e7e3fc94c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.672ex; height:3.343ex;" alt="{\displaystyle \Delta (x)=\left\vert \psi (x)-x\right\vert <{\sqrt {x}}\log ^{2}(x)/(8\pi )}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>73.2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mn>73.2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>73.2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef578699eb4f539a0e14ad934937984102ff80c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.562ex; height:2.176ex;" alt="{\displaystyle x>73.2}"></span>. If we analyze the Chebyshev deviations Δ(<i>n</i>) on the integers <i>n</i> using the method of expanding bins and plot the variance versus the mean a variance to mean power law can be demonstrated.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2019)">citation needed</span></a></i>]</sup> Moreover, these deviations correspond to the Tweedie compound Poisson-gamma distribution and they exhibit 1/<i>f</i> noise. </p> <div class="mw-heading mw-heading3"><h3 id="Other_applications">Other applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=20" title="Edit section: Other applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Applications of Tweedie distributions include: </p> <ul><li>actuarial studies<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup></li> <li>assay analysis <sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup></li> <li>survival analysis<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup></li> <li>ecology <sup id="cite_ref-Kendal2002_13-2" class="reference"><a href="#cite_note-Kendal2002-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li> <li>analysis of alcohol consumption in British teenagers <sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup></li> <li>medical applications <sup id="cite_ref-smyth1996_51-0" class="reference"><a href="#cite_note-smyth1996-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup></li> <li>health economics <sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup></li> <li>meteorology and climatology <sup id="cite_ref-smyth1996_51-1" class="reference"><a href="#cite_note-smyth1996-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup></li> <li>fisheries <sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Mertens_function" title="Mertens function">Mertens function</a><sup id="cite_ref-Kendal2011a_55-0" class="reference"><a href="#cite_note-Kendal2011a-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Self-organized_criticality" title="Self-organized criticality">self-organized criticality</a><sup id="cite_ref-Kendal2015_56-0" class="reference"><a href="#cite_note-Kendal2015-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tweedie_distribution&action=edit&section=21" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 40em;"> <ol class="references"> <li id="cite_note-t84-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-t84_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-t84_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFTweedie1984" class="citation conference cs1">Tweedie, M.C.K. (1984). "An index which distinguishes between some important exponential families". In Ghosh, J.K.; Roy, J (eds.). <i>Statistics: Applications and New Directions</i>. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference. Calcutta: Indian Statistical Institute. pp. 579–604. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0786162">0786162</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=An+index+which+distinguishes+between+some+important+exponential+families&rft.btitle=Statistics%3A+Applications+and+New+Directions&rft.place=Calcutta&rft.pages=579-604&rft.pub=Indian+Statistical+Institute&rft.date=1984&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D786162%23id-name%3DMR&rft.aulast=Tweedie&rft.aufirst=M.C.K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATweedie+distribution" class="Z3988"></span></span> </li> <li id="cite_note-Jørgensen-1997-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Jørgensen-1997_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Jørgensen-1997_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Jørgensen-1997_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Jørgensen-1997_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Jørgensen-1997_2-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJørgensen1997" class="citation book cs1">Jørgensen, Bent (1997). <i>The theory of dispersion models</i>. Chapman & Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0412997112" title="Special:BookSources/978-0412997112"><bdi>978-0412997112</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+theory+of+dispersion+models&rft.pub=Chapman+%26+Hall&rft.date=1997&rft.isbn=978-0412997112&rft.aulast=J%C3%B8rgensen&rft.aufirst=Bent&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATweedie+distribution" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJørgensen,_B1987" class="citation journal cs1">Jørgensen, B (1987). "Exponential dispersion models". <i>Journal of the Royal Statistical Society, Series B</i>. <b>49</b> (2): 127–162. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2345415">2345415</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+B&rft.atitle=Exponential+dispersion+models&rft.volume=49&rft.issue=2&rft.pages=127-162&rft.date=1987&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2345415%23id-name%3DJSTOR&rft.au=J%C3%B8rgensen%2C+B&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATweedie+distribution" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith1997" class="citation journal cs1"><a href="/wiki/Cedric_Smith_(statistician)" title="Cedric Smith (statistician)">Smith, C.A.B.</a> (1997). <a rel="nofollow" class="external text" href="https://doi.org/10.1111%2F1467-985X.00052">"Obituary: Maurice Charles Kenneth Tweedie, 1919–96"</a>. <i><a href="/wiki/Journal_of_the_Royal_Statistical_Society,_Series_A" class="mw-redirect" title="Journal of the Royal Statistical Society, Series A">Journal of the Royal Statistical Society, Series A</a></i>. <b>160</b> (1): 151–154. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1111%2F1467-985X.00052">10.1111/1467-985X.00052</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+A&rft.atitle=Obituary%3A+Maurice+Charles+Kenneth+Tweedie%2C+1919%E2%80%9396&rft.volume=160&rft.issue=1&rft.pages=151-154&rft.date=1997&rft_id=info%3Adoi%2F10.1111%2F1467-985X.00052&rft.aulast=Smith&rft.aufirst=C.A.B.&rft_id=https%3A%2F%2Fdoi.org%2F10.1111%252F1467-985X.00052&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATweedie+distribution" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBar-LevEnis1985" class="citation journal cs1">Bar-Lev, S.K.; Enis, P. (1985). "Reproducibility in the one-parameter exponential family". <i><a href="/wiki/Metrika" class="mw-redirect" title="Metrika">Metrika</a></i>. <b>32</b> (1): 391–394. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01897827">10.1007/BF01897827</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Metrika&rft.atitle=Reproducibility+in+the+one-parameter+exponential+family&rft.volume=32&rft.issue=1&rft.pages=391-394&rft.date=1985&rft_id=info%3Adoi%2F10.1007%2FBF01897827&rft.aulast=Bar-Lev&rft.aufirst=S.K.&rft.au=Enis%2C+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATweedie+distribution" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBar-LevEnis1986" class="citation journal cs1">Bar-Lev, Shall.K.; Enis, Peter (1986). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176350173">"Reproducibility and Natural Exponential Families with Power Variance Functions"</a>. <i><a href="/wiki/The_Annals_of_Statistics" class="mw-redirect" title="The Annals of Statistics">The Annals of Statistics</a></i>. <b>14</b> (4): 1507–1522. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176350173">10.1214/aos/1176350173</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Statistics&rft.atitle=Reproducibility+and+Natural+Exponential+Families+with+Power+Variance+Functions&rft.volume=14&rft.issue=4&rft.pages=1507-1522&rft.date=1986&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176350173&rft.aulast=Bar-Lev&rft.aufirst=Shall.K.&rft.au=Enis%2C+Peter&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176350173&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATweedie+distribution" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJørgensen,_BMartinez,_JRTsao,_M1994" class="citation journal cs1">Jørgensen, B; Martinez, JR; Tsao, M (1994). "Asymptotic behaviour of the variance function". <i>Scandinavian Journal of Statistics</i>. <b>21</b>: 223–243.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scandinavian+Journal+of+Statistics&rft.atitle=Asymptotic+behaviour+of+the+variance+function&rft.volume=21&rft.pages=223-243&rft.date=1994&rft.au=J%C3%B8rgensen%2C+B&rft.au=Martinez%2C+JR&rft.au=Tsao%2C+M&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATweedie+distribution" class="Z3988"></span></span> </li> <li id="cite_note-Kendal2011b-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kendal2011b_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kendal2011b_8-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Kendal2011b_8-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Kendal2011b_8-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Kendal2011b_8-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Kendal2011b_8-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Kendal2011b_8-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-Kendal2011b_8-7"><sup><i><b>h</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKendalJørgensen2011" class="citation journal cs1">Kendal, W. S.; Jørgensen, B. 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New Series. <b>7</b>: 160–165.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Virginia+J.+Sci.&rft.atitle=Some+statistical+properties+of+Inverse+Gaussian+distributions&rft.volume=7&rft.pages=160-165&rft.date=1956&rft.aulast=Tweedie&rft.aufirst=M.C.K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATweedie+distribution" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol 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style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Probability_distributions" title="Template:Probability distributions"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Probability_distributions" title="Template talk:Probability distributions"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_distributions" title="Special:EditPage/Template:Probability distributions"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Probability_distributions_(list)" style="font-size:114%;margin:0 4em"><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distributions</a> (<a href="/wiki/List_of_probability_distributions" title="List of probability distributions">list</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Discrete <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">with finite <br />support</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benford%27s_law" title="Benford's law">Benford</a></li> <li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli</a></li> <li><a href="/wiki/Beta-binomial_distribution" title="Beta-binomial distribution">Beta-binomial</a></li> <li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial</a></li> <li><a href="/wiki/Categorical_distribution" title="Categorical distribution">Categorical</a></li> <li><a href="/wiki/Hypergeometric_distribution" title="Hypergeometric distribution">Hypergeometric</a> <ul><li><a href="/wiki/Negative_hypergeometric_distribution" title="Negative hypergeometric distribution">Negative</a></li></ul></li> <li><a href="/wiki/Poisson_binomial_distribution" title="Poisson binomial distribution">Poisson binomial</a></li> <li><a href="/wiki/Rademacher_distribution" title="Rademacher distribution">Rademacher</a></li> <li><a href="/wiki/Soliton_distribution" title="Soliton distribution">Soliton</a></li> <li><a href="/wiki/Discrete_uniform_distribution" title="Discrete uniform distribution">Discrete uniform</a></li> <li><a href="/wiki/Zipf%27s_law" title="Zipf's law">Zipf</a></li> <li><a href="/wiki/Zipf%E2%80%93Mandelbrot_law" title="Zipf–Mandelbrot law">Zipf–Mandelbrot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with infinite <br />support</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Beta_negative_binomial_distribution" title="Beta negative binomial distribution">Beta negative binomial</a></li> <li><a href="/wiki/Borel_distribution" title="Borel distribution">Borel</a></li> <li><a href="/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution" title="Conway–Maxwell–Poisson distribution">Conway–Maxwell–Poisson</a></li> <li><a href="/wiki/Discrete_phase-type_distribution" title="Discrete phase-type distribution">Discrete phase-type</a></li> <li><a href="/wiki/Delaporte_distribution" title="Delaporte distribution">Delaporte</a></li> <li><a href="/wiki/Extended_negative_binomial_distribution" title="Extended negative binomial distribution">Extended negative binomial</a></li> <li><a href="/wiki/Flory%E2%80%93Schulz_distribution" title="Flory–Schulz distribution">Flory–Schulz</a></li> <li><a href="/wiki/Gauss%E2%80%93Kuzmin_distribution" title="Gauss–Kuzmin distribution">Gauss–Kuzmin</a></li> <li><a href="/wiki/Geometric_distribution" title="Geometric distribution">Geometric</a></li> <li><a href="/wiki/Logarithmic_distribution" title="Logarithmic distribution">Logarithmic</a></li> <li><a href="/wiki/Mixed_Poisson_distribution" title="Mixed Poisson distribution">Mixed Poisson</a></li> <li><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">Negative binomial</a></li> <li><a href="/wiki/(a,b,0)_class_of_distributions" title="(a,b,0) class of distributions">Panjer</a></li> <li><a href="/wiki/Parabolic_fractal_distribution" title="Parabolic fractal distribution">Parabolic fractal</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a></li> <li><a href="/wiki/Skellam_distribution" title="Skellam distribution">Skellam</a></li> <li><a href="/wiki/Yule%E2%80%93Simon_distribution" title="Yule–Simon distribution">Yule–Simon</a></li> <li><a href="/wiki/Zeta_distribution" title="Zeta distribution">Zeta</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Continuous <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">supported on a <br />bounded interval</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arcsine_distribution" title="Arcsine distribution">Arcsine</a></li> <li><a href="/wiki/ARGUS_distribution" title="ARGUS distribution">ARGUS</a></li> <li><a href="/wiki/Balding%E2%80%93Nichols_model" title="Balding–Nichols model">Balding–Nichols</a></li> <li><a href="/wiki/Bates_distribution" title="Bates distribution">Bates</a></li> <li><a href="/wiki/Beta_distribution" title="Beta distribution">Beta</a> <ul><li><a href="/wiki/Generalized_beta_distribution" title="Generalized beta distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Beta_rectangular_distribution" title="Beta rectangular distribution">Beta rectangular</a></li> <li><a href="/wiki/Continuous_Bernoulli_distribution" title="Continuous Bernoulli distribution">Continuous Bernoulli</a></li> <li><a href="/wiki/Irwin%E2%80%93Hall_distribution" title="Irwin–Hall distribution">Irwin–Hall</a></li> <li><a href="/wiki/Kumaraswamy_distribution" title="Kumaraswamy distribution">Kumaraswamy</a></li> <li><a href="/wiki/Logit-normal_distribution" title="Logit-normal distribution">Logit-normal</a></li> <li><a href="/wiki/Noncentral_beta_distribution" title="Noncentral beta distribution">Noncentral beta</a></li> <li><a href="/wiki/PERT_distribution" title="PERT distribution">PERT</a></li> <li><a href="/wiki/Raised_cosine_distribution" title="Raised cosine distribution">Raised cosine</a></li> <li><a href="/wiki/Reciprocal_distribution" title="Reciprocal distribution">Reciprocal</a></li> <li><a href="/wiki/Triangular_distribution" title="Triangular distribution">Triangular</a></li> <li><a href="/wiki/U-quadratic_distribution" title="U-quadratic distribution">U-quadratic</a></li> <li><a href="/wiki/Continuous_uniform_distribution" title="Continuous uniform distribution">Uniform</a></li> <li><a href="/wiki/Wigner_semicircle_distribution" title="Wigner semicircle distribution">Wigner semicircle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported on a <br />semi-infinite <br />interval</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benini_distribution" title="Benini distribution">Benini</a></li> <li><a href="/wiki/Benktander_type_I_distribution" title="Benktander type I distribution">Benktander 1st kind</a></li> <li><a href="/wiki/Benktander_type_II_distribution" title="Benktander type II distribution">Benktander 2nd kind</a></li> <li><a href="/wiki/Beta_prime_distribution" title="Beta prime distribution">Beta prime</a></li> <li><a href="/wiki/Burr_distribution" title="Burr distribution">Burr</a></li> <li><a href="/wiki/Chi_distribution" title="Chi distribution">Chi</a></li> <li><a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">Chi-squared</a> <ul><li><a href="/wiki/Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">Noncentral</a></li> <li><a href="/wiki/Inverse-chi-squared_distribution" title="Inverse-chi-squared distribution">Inverse</a> <ul><li><a href="/wiki/Scaled_inverse_chi-squared_distribution" title="Scaled inverse chi-squared distribution">Scaled</a></li></ul></li></ul></li> <li><a href="/wiki/Dagum_distribution" title="Dagum distribution">Dagum</a></li> <li><a href="/wiki/Davis_distribution" title="Davis distribution">Davis</a></li> <li><a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang</a> <ul><li><a href="/wiki/Hyper-Erlang_distribution" title="Hyper-Erlang distribution">Hyper</a></li></ul></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential</a> <ul><li><a href="/wiki/Hyperexponential_distribution" title="Hyperexponential distribution">Hyperexponential</a></li> <li><a href="/wiki/Hypoexponential_distribution" title="Hypoexponential distribution">Hypoexponential</a></li> <li><a href="/wiki/Exponential-logarithmic_distribution" title="Exponential-logarithmic distribution">Logarithmic</a></li></ul></li> <li><a href="/wiki/F-distribution" title="F-distribution"><i>F</i></a> <ul><li><a href="/wiki/Noncentral_F-distribution" title="Noncentral F-distribution">Noncentral</a></li></ul></li> <li><a href="/wiki/Folded_normal_distribution" title="Folded normal distribution">Folded normal</a></li> <li><a href="/wiki/Fr%C3%A9chet_distribution" title="Fréchet distribution">Fréchet</a></li> <li><a href="/wiki/Gamma_distribution" title="Gamma distribution">Gamma</a> <ul><li><a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">Generalized</a></li> <li><a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Gamma/Gompertz_distribution" title="Gamma/Gompertz distribution">gamma/Gompertz</a></li> <li><a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz</a> <ul><li><a href="/wiki/Shifted_Gompertz_distribution" title="Shifted Gompertz distribution">Shifted</a></li></ul></li> <li><a href="/wiki/Half-logistic_distribution" title="Half-logistic distribution">Half-logistic</a></li> <li><a href="/wiki/Half-normal_distribution" title="Half-normal distribution">Half-normal</a></li> <li><a href="/wiki/Hotelling%27s_T-squared_distribution" title="Hotelling's T-squared distribution">Hotelling's <i>T</i>-squared</a></li> <li><a href="/wiki/Inverse_Gaussian_distribution" title="Inverse Gaussian distribution">Inverse Gaussian</a> <ul><li><a href="/wiki/Generalized_inverse_Gaussian_distribution" title="Generalized inverse Gaussian distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov</a></li> <li><a href="/wiki/L%C3%A9vy_distribution" title="Lévy distribution">Lévy</a></li> <li><a href="/wiki/Log-Cauchy_distribution" title="Log-Cauchy distribution">Log-Cauchy</a></li> <li><a href="/wiki/Log-Laplace_distribution" title="Log-Laplace distribution">Log-Laplace</a></li> <li><a href="/wiki/Log-logistic_distribution" title="Log-logistic distribution">Log-logistic</a></li> <li><a href="/wiki/Log-normal_distribution" title="Log-normal distribution">Log-normal</a></li> <li><a href="/wiki/Log-t_distribution" title="Log-t distribution">Log-t</a></li> <li><a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax</a></li> <li><a href="/wiki/Matrix-exponential_distribution" title="Matrix-exponential distribution">Matrix-exponential</a></li> <li><a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann</a></li> <li><a href="/wiki/Maxwell%E2%80%93J%C3%BCttner_distribution" title="Maxwell–Jüttner distribution">Maxwell–Jüttner</a></li> <li><a href="/wiki/Mittag-Leffler_distribution" title="Mittag-Leffler distribution">Mittag-Leffler</a></li> <li><a href="/wiki/Nakagami_distribution" title="Nakagami distribution">Nakagami</a></li> <li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto</a></li> <li><a href="/wiki/Phase-type_distribution" title="Phase-type distribution">Phase-type</a></li> <li><a href="/wiki/Poly-Weibull_distribution" title="Poly-Weibull distribution">Poly-Weibull</a></li> <li><a href="/wiki/Rayleigh_distribution" title="Rayleigh distribution">Rayleigh</a></li> <li><a href="/wiki/Relativistic_Breit%E2%80%93Wigner_distribution" title="Relativistic Breit–Wigner distribution">Relativistic Breit–Wigner</a></li> <li><a href="/wiki/Rice_distribution" title="Rice distribution">Rice</a></li> <li><a href="/wiki/Truncated_normal_distribution" title="Truncated normal distribution">Truncated normal</a></li> <li><a href="/wiki/Type-2_Gumbel_distribution" title="Type-2 Gumbel distribution">type-2 Gumbel</a></li> <li><a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull</a> <ul><li><a href="/wiki/Discrete_Weibull_distribution" title="Discrete Weibull distribution">Discrete</a></li></ul></li> <li><a href="/wiki/Wilks%27s_lambda_distribution" title="Wilks's lambda distribution">Wilks's lambda</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported <br />on the whole <br />real line</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy</a></li> <li><a href="/wiki/Generalized_normal_distribution#Version_1" title="Generalized normal distribution">Exponential power</a></li> <li><a href="/wiki/Fisher%27s_z-distribution" title="Fisher's z-distribution">Fisher's <i>z</i></a></li> <li><a href="/wiki/Kaniadakis_Gaussian_distribution" title="Kaniadakis Gaussian distribution">Kaniadakis κ-Gaussian</a></li> <li><a href="/wiki/Gaussian_q-distribution" title="Gaussian q-distribution">Gaussian <i>q</i></a></li> <li><a href="/wiki/Generalized_normal_distribution" title="Generalized normal distribution">Generalized normal</a></li> <li><a href="/wiki/Generalised_hyperbolic_distribution" title="Generalised hyperbolic distribution">Generalized hyperbolic</a></li> <li><a href="/wiki/Geometric_stable_distribution" title="Geometric stable distribution">Geometric stable</a></li> <li><a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel</a></li> <li><a href="/wiki/Holtsmark_distribution" title="Holtsmark distribution">Holtsmark</a></li> <li><a href="/wiki/Hyperbolic_secant_distribution" title="Hyperbolic secant distribution">Hyperbolic secant</a></li> <li><a href="/wiki/Johnson%27s_SU-distribution" title="Johnson's SU-distribution">Johnson's <i>S<sub>U</sub></i></a></li> <li><a href="/wiki/Landau_distribution" title="Landau distribution">Landau</a></li> <li><a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace</a> <ul><li><a href="/wiki/Asymmetric_Laplace_distribution" title="Asymmetric Laplace distribution">Asymmetric</a></li></ul></li> <li><a href="/wiki/Logistic_distribution" title="Logistic distribution">Logistic</a></li> <li><a href="/wiki/Noncentral_t-distribution" title="Noncentral t-distribution">Noncentral <i>t</i></a></li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">Normal (Gaussian)</a></li> <li><a href="/wiki/Normal-inverse_Gaussian_distribution" title="Normal-inverse Gaussian distribution">Normal-inverse Gaussian</a></li> <li><a href="/wiki/Skew_normal_distribution" title="Skew normal distribution">Skew normal</a></li> <li><a href="/wiki/Slash_distribution" title="Slash distribution">Slash</a></li> <li><a href="/wiki/Stable_distribution" title="Stable distribution">Stable</a></li> <li><a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's <i>t</i></a></li> <li><a href="/wiki/Tracy%E2%80%93Widom_distribution" title="Tracy–Widom distribution">Tracy–Widom</a></li> <li><a href="/wiki/Variance-gamma_distribution" title="Variance-gamma distribution">Variance-gamma</a></li> <li><a href="/wiki/Voigt_profile" title="Voigt profile">Voigt</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with support <br />whose type varies</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Generalized_chi-squared_distribution" title="Generalized chi-squared distribution">Generalized chi-squared</a></li> <li><a href="/wiki/Generalized_extreme_value_distribution" title="Generalized extreme value distribution">Generalized extreme value</a></li> <li><a href="/wiki/Generalized_Pareto_distribution" title="Generalized Pareto distribution">Generalized Pareto</a></li> <li><a href="/wiki/Marchenko%E2%80%93Pastur_distribution" title="Marchenko–Pastur distribution">Marchenko–Pastur</a></li> <li><a href="/wiki/Kaniadakis_Exponential_distribution" class="mw-redirect" title="Kaniadakis Exponential distribution">Kaniadakis <i>κ</i>-exponential</a></li> <li><a href="/wiki/Kaniadakis_Gamma_distribution" title="Kaniadakis Gamma distribution">Kaniadakis <i>κ</i>-Gamma</a></li> <li><a href="/wiki/Kaniadakis_Weibull_distribution" title="Kaniadakis Weibull distribution">Kaniadakis <i>κ</i>-Weibull</a></li> <li><a href="/wiki/Kaniadakis_Logistic_distribution" class="mw-redirect" title="Kaniadakis Logistic distribution">Kaniadakis <i>κ</i>-Logistic</a></li> <li><a href="/wiki/Kaniadakis_Erlang_distribution" title="Kaniadakis Erlang distribution">Kaniadakis <i>κ</i>-Erlang</a></li> <li><a href="/wiki/Q-exponential_distribution" title="Q-exponential distribution"><i>q</i>-exponential</a></li> <li><a href="/wiki/Q-Gaussian_distribution" title="Q-Gaussian distribution"><i>q</i>-Gaussian</a></li> <li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution"><i>q</i>-Weibull</a></li> <li><a href="/wiki/Shifted_log-logistic_distribution" title="Shifted log-logistic distribution">Shifted log-logistic</a></li> <li><a href="/wiki/Tukey_lambda_distribution" title="Tukey lambda distribution">Tukey lambda</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mixed <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">continuous-<br />discrete</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Rectified_Gaussian_distribution" title="Rectified Gaussian distribution">Rectified Gaussian</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Multivariate <br />(joint)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="nobold"><i>Discrete: </i></span></li> <li><a href="/wiki/Ewens%27s_sampling_formula" title="Ewens's sampling formula">Ewens</a></li> <li><a href="/wiki/Multinomial_distribution" title="Multinomial distribution">Multinomial</a> <ul><li><a href="/wiki/Dirichlet-multinomial_distribution" title="Dirichlet-multinomial distribution">Dirichlet</a></li> <li><a href="/wiki/Negative_multinomial_distribution" title="Negative multinomial distribution">Negative</a></li></ul></li> <li><span class="nobold"><i>Continuous: </i></span></li> <li><a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet</a> <ul><li><a href="/wiki/Generalized_Dirichlet_distribution" title="Generalized Dirichlet distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Multivariate_Laplace_distribution" title="Multivariate Laplace distribution">Multivariate Laplace</a></li> <li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Multivariate normal</a></li> <li><a href="/wiki/Multivariate_stable_distribution" title="Multivariate stable distribution">Multivariate stable</a></li> <li><a href="/wiki/Multivariate_t-distribution" title="Multivariate t-distribution">Multivariate <i>t</i></a></li> <li><a href="/wiki/Normal-gamma_distribution" title="Normal-gamma distribution">Normal-gamma</a> <ul><li><a href="/wiki/Normal-inverse-gamma_distribution" title="Normal-inverse-gamma distribution">Inverse</a></li></ul></li> <li><span class="nobold"><i><a href="/wiki/Random_matrix" title="Random matrix">Matrix-valued: </a></i></span></li> <li><a href="/wiki/Lewandowski-Kurowicka-Joe_distribution" title="Lewandowski-Kurowicka-Joe distribution">LKJ</a></li> <li><a href="/wiki/Matrix_normal_distribution" title="Matrix normal distribution">Matrix normal</a></li> <li><a href="/wiki/Matrix_t-distribution" title="Matrix t-distribution">Matrix <i>t</i></a></li> <li><a href="/wiki/Matrix_gamma_distribution" title="Matrix gamma distribution">Matrix gamma</a> <ul><li><a href="/wiki/Inverse_matrix_gamma_distribution" title="Inverse matrix gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Wishart_distribution" title="Wishart distribution">Wishart</a> <ul><li><a href="/wiki/Normal-Wishart_distribution" title="Normal-Wishart distribution">Normal</a></li> <li><a href="/wiki/Inverse-Wishart_distribution" title="Inverse-Wishart distribution">Inverse</a></li> <li><a href="/wiki/Normal-inverse-Wishart_distribution" title="Normal-inverse-Wishart distribution">Normal-inverse</a></li> <li><a href="/wiki/Complex_Wishart_distribution" title="Complex Wishart distribution">Complex</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Directional_statistics" title="Directional statistics">Directional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><span class="nobold"><i>Univariate (circular) <a href="/wiki/Directional_statistics" title="Directional statistics">directional</a></i></span></dt> <dd><a href="/wiki/Circular_uniform_distribution" title="Circular uniform distribution">Circular uniform</a></dd> <dd><a href="/wiki/Von_Mises_distribution" title="Von Mises distribution">Univariate von Mises</a></dd> <dd><a href="/wiki/Wrapped_normal_distribution" title="Wrapped normal distribution">Wrapped normal</a></dd> <dd><a href="/wiki/Wrapped_Cauchy_distribution" title="Wrapped Cauchy distribution">Wrapped Cauchy</a></dd> <dd><a href="/wiki/Wrapped_exponential_distribution" title="Wrapped exponential distribution">Wrapped exponential</a></dd> <dd><a href="/wiki/Wrapped_asymmetric_Laplace_distribution" title="Wrapped asymmetric Laplace distribution">Wrapped asymmetric Laplace</a></dd> <dd><a href="/wiki/Wrapped_L%C3%A9vy_distribution" title="Wrapped Lévy distribution">Wrapped Lévy</a></dd> <dt><span class="nobold"><i>Bivariate (spherical)</i></span></dt> <dd><a href="/wiki/Kent_distribution" title="Kent distribution">Kent</a></dd> <dt><span class="nobold"><i>Bivariate (toroidal)</i></span></dt> <dd><a href="/wiki/Bivariate_von_Mises_distribution" title="Bivariate von Mises distribution">Bivariate von Mises</a></dd> <dt><span class="nobold"><i>Multivariate</i></span></dt> <dd><a href="/wiki/Von_Mises%E2%80%93Fisher_distribution" title="Von Mises–Fisher distribution">von Mises–Fisher</a></dd> <dd><a href="/wiki/Bingham_distribution" title="Bingham distribution">Bingham</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Degenerate_distribution" title="Degenerate distribution">Degenerate</a> <br />and <a href="/wiki/Singular_distribution" title="Singular distribution">singular</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><span class="nobold"><i>Degenerate</i></span></dt> <dd><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></dd> <dt><span class="nobold"><i>Singular</i></span></dt> <dd><a href="/wiki/Cantor_distribution" title="Cantor distribution">Cantor</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Families</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Circular_distribution" title="Circular distribution">Circular</a></li> <li><a href="/wiki/Compound_Poisson_distribution" title="Compound Poisson distribution">Compound Poisson</a></li> <li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential</a></li> <li><a href="/wiki/Natural_exponential_family" title="Natural exponential family">Natural exponential</a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale</a></li> <li><a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">Maximum entropy</a></li> <li><a href="/wiki/Mixture_distribution" title="Mixture distribution">Mixture</a></li> <li><a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson</a></li> <li><a class="mw-selflink selflink">Tweedie</a></li> <li><a href="/wiki/Wrapped_distribution" title="Wrapped distribution">Wrapped</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" 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