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class="vector-toc-list"> <li id="toc-Discrete_Weibull_vs._Poisson_Distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discrete_Weibull_vs._Poisson_Distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Discrete Weibull vs. Poisson Distribution</span> </div> </a> <ul id="toc-Discrete_Weibull_vs._Poisson_Distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discrete_Weibull_vs._Geometric_Distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discrete_Weibull_vs._Geometric_Distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Discrete Weibull vs. Geometric Distribution</span> </div> </a> <ul id="toc-Discrete_Weibull_vs._Geometric_Distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discrete_Weibull_vs._Negative_Binomial_Distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discrete_Weibull_vs._Negative_Binomial_Distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Discrete Weibull vs. Negative Binomial Distribution</span> </div> </a> <ul id="toc-Discrete_Weibull_vs._Negative_Binomial_Distribution-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> 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width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}"></span> <a href="/wiki/Scale_parameter" title="Scale parameter">scale</a> <br /><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 \beta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a87dc52878418173659e6d0ff8e77ab2897eac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta >0}"></span> <a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Support_(mathematics)" title="Support (mathematics)">Support</a></th><td colspan="3" class="infobox-data"> <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\in \{0,1,2,\ldots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \{0,1,2,\ldots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/826c18832db127cedd25a5709af4cdea562f594a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.808ex; height:2.843ex;" alt="{\displaystyle x\in \{0,1,2,\ldots \}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability_mass_function" title="Probability mass function">PMF</a></th><td colspan="3" class="infobox-data"> <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 \exp \left[-\left({\frac {x}{\alpha }}\right)^{\beta }\right]-\exp \left[-\left({\frac {x+1}{\alpha }}\right)^{\beta }\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>α</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>α</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \left[-\left({\frac {x}{\alpha }}\right)^{\beta }\right]-\exp \left[-\left({\frac {x+1}{\alpha }}\right)^{\beta }\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/108e0877183ce48d9a98ec953cbff9c6489ca272" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.764ex; height:7.509ex;" alt="{\displaystyle \exp \left[-\left({\frac {x}{\alpha }}\right)^{\beta }\right]-\exp \left[-\left({\frac {x+1}{\alpha }}\right)^{\beta }\right]}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">CDF</a></th><td colspan="3" class="infobox-data"> <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 1-\exp \left[-\left({\frac {x+1}{\alpha }}\right)^{\beta }\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>α</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\exp \left[-\left({\frac {x+1}{\alpha }}\right)^{\beta }\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70736d03d8cf3850c4325764905bba39bc3c50c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.838ex; height:7.509ex;" alt="{\displaystyle 1-\exp \left[-\left({\frac {x+1}{\alpha }}\right)^{\beta }\right]}"></span></td></tr></tbody></table> <p>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, the <b>discrete Weibull distribution</b> is the <a href="/wiki/Probability_distribution#Discrete_probability_distribution" title="Probability distribution">discrete</a> variant of the <a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull distribution</a>. The Discrete Weibull Distribution, first introduced by Toshio Nakagawa and Shunji Osaki, is a discrete analog of the continuous Weibull distribution, predominantly used in reliability engineering. It is particularly applicable for modeling failure data measured in discrete units like cycles or shocks. This distribution provides a versatile tool for analyzing scenarios where the timing of events is counted in distinct intervals, making it distinctively useful in fields that deal with discrete data patterns and reliability analysis. The discrete Weibull distribution is <a href="/wiki/Infinite_divisibility_(probability)" title="Infinite divisibility (probability)">infinitely divisible</a> only for <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 0<\beta \leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>β</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\beta \leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7424a8247936a47fd9cb801f5bac8b5a1fcda817" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.854ex; height:2.509ex;" alt="{\displaystyle 0<\beta \leq 1}"></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Alternative_parametrizations">Alternative parametrizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_Weibull_distribution&action=edit&section=1" title="Edit section: Alternative parametrizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> In the original paper by Nakagawa and Osaki they used the parametrization <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=q^{k^{-\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=q^{k^{-\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8852e07fe481ec792ec51dded5b39bbe3f855ff7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.492ex; height:3.509ex;" alt="{\displaystyle p=q^{k^{-\beta }}}"></span> making the cumulative distribution 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 1-q^{(x+1)^{\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-q^{(x+1)^{\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91234c6334c7a1ad392556d6cd81364d0b098b21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.564ex; height:3.509ex;" alt="{\displaystyle 1-q^{(x+1)^{\beta }}}"></span> </p><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:CDF-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/CDF-2.png/220px-CDF-2.png" decoding="async" width="220" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/CDF-2.png/330px-CDF-2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/CDF-2.png/440px-CDF-2.png 2x" data-file-width="720" data-file-height="530" /></a><figcaption>The CDF of the Discrete Weibull Distribution with a q value of 0.5 and k values of 1 through 5. The B values are as follows: Red = 0.5, Green = 1.0, Blue = 1.5, Purple = 2.0, Orange = 2.5.</figcaption></figure><p> 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 q\in (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\in (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3d535fcce8ea6a653bb201b611657c62c00f4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.078ex; height:2.843ex;" alt="{\displaystyle q\in (0,1)}"></span> and the probability mass 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 q^{k^{-\beta }}-q^{(k+1)^{\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> </mrow> </msup> </mrow> </msup> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{k^{-\beta }}-q^{(k+1)^{\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b7c2558f77f2276eef43c1e19543bed6d58e6a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.452ex; height:3.509ex;" alt="{\displaystyle q^{k^{-\beta }}-q^{(k+1)^{\beta }}}"></span> </p><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:PMF-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/PMF-2.png/220px-PMF-2.png" decoding="async" width="220" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/PMF-2.png/330px-PMF-2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/PMF-2.png/440px-PMF-2.png 2x" data-file-width="720" data-file-height="534" /></a><figcaption>The PMF of the Discrete Weibull Distribution with a q value of 0.5 and k values of 1 through 5. The B values are as follows: Red = 0.5, Green = 1.0, Blue = 1.5, Purple = 2.0, Orange = 2.5.</figcaption></figure> <p>. Setting <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 \beta =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73416922785589e358ae2bb10c7633667b4c24a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta =1}"></span> makes the relationship with the geometric distribution apparent.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>An alternative parametrization — related to the <a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a> — has been used to estimate parameters in <a href="/wiki/Infectious_disease_modelling" class="mw-redirect" title="Infectious disease modelling">infectious disease modelling</a>.<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> This parametrization introduces a 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 \kappa ={\frac {\beta }{\alpha ^{\beta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa ={\frac {\beta }{\alpha ^{\beta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bb6929791a274e09a58b46410e6806327327a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.935ex; height:5.676ex;" alt="{\displaystyle \kappa ={\frac {\beta }{\alpha ^{\beta }}}}"></span>, meaning that the term <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 \left({\frac {1}{\alpha }}\right)^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {1}{\alpha }}\right)^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bdc1e0e4638af1787de7d27ad9151f866f10f47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:6.919ex; height:6.676ex;" alt="{\displaystyle \left({\frac {1}{\alpha }}\right)^{\beta }}"></span> can be replaced 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 {\frac {\kappa }{\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>κ</mi> <mi>β</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\kappa }{\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a084b4894d2b0fbb131fd129ce103f1c1957bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.175ex; height:5.176ex;" alt="{\displaystyle {\frac {\kappa }{\beta }}}"></span>. Therefore, the probability mass function can be expressed as </p> <dl><dd><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 \exp \left[-{\frac {\kappa x^{\beta }}{\beta }}\right]-\exp \left[-{\frac {\kappa \left(x+1\right)^{\beta }}{\beta }}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>κ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mi>β</mi> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>κ</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mi>β</mi> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \left[-{\frac {\kappa x^{\beta }}{\beta }}\right]-\exp \left[-{\frac {\kappa \left(x+1\right)^{\beta }}{\beta }}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a818181f214472f0f1391a8fdd39c45634c58f59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.897ex; height:7.509ex;" alt="{\displaystyle \exp \left[-{\frac {\kappa x^{\beta }}{\beta }}\right]-\exp \left[-{\frac {\kappa \left(x+1\right)^{\beta }}{\beta }}\right]}"></span>,</dd></dl> <p>and the cumulative mass function can be expressed as </p> <dl><dd><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 1-\exp \left[-{\frac {\kappa \left(x+1\right)^{\beta }}{\beta }}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>κ</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mi>β</mi> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\exp \left[-{\frac {\kappa \left(x+1\right)^{\beta }}{\beta }}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/985cc08c75b6f2a06abf370b37102603bf00143d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.565ex; height:7.509ex;" alt="{\displaystyle 1-\exp \left[-{\frac {\kappa \left(x+1\right)^{\beta }}{\beta }}\right]}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Location-scale_transformation">Location-scale transformation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_Weibull_distribution&action=edit&section=2" title="Edit section: Location-scale transformation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The continuous Weibull distribution has a close relationship with the <a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel distribution</a> which is easy to see when log-transforming the variable. A similar transformation can be made on the discrete Weibull. </p><p>Define <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 e^{Y}-1=X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{Y}-1=X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c891851e698f8942746a936e509b297b5b260e72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.651ex; height:2.843ex;" alt="{\displaystyle e^{Y}-1=X}"></span> where (unconventionally) <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=\log(X+1)\in \{\log(1),\log(2),\ldots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=\log(X+1)\in \{\log(1),\log(2),\ldots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3766e176ce6d1d07d338a0f75f55def75a147336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.479ex; height:2.843ex;" alt="{\displaystyle Y=\log(X+1)\in \{\log(1),\log(2),\ldots \}}"></span> and define parameters <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 =\log(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\log(\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/363b31f79141798ae6e411759568e80955f80d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.769ex; height:2.843ex;" alt="{\displaystyle \mu =\log(\alpha )}"></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 ={\frac {1}{\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>β</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\frac {1}{\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/735a0c7e503488b68a4768c891e55412aaab5785" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.596ex; height:5.676ex;" alt="{\displaystyle \sigma ={\frac {1}{\beta }}}"></span>. By replacing <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> in the cumulative mass function: </p> <dl><dd><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 \Pr(X\leq x)=\Pr(X\leq e^{y}-1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(X\leq x)=\Pr(X\leq e^{y}-1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a513ba75a3648acf8bec93045e9e55a8c52ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.975ex; height:2.843ex;" alt="{\displaystyle \Pr(X\leq x)=\Pr(X\leq e^{y}-1).}"></span></dd></dl> <p>We see that we get a location-scale parametrization: </p> <dl><dd><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 =1-\exp \left[-\left({\frac {x+1}{\alpha }}\right)^{\beta }\right]=1-\exp \left[-\left({\frac {e^{y}}{e^{\mu }}}\right)^{\frac {1}{\sigma }}\right]=1-\exp \left[-\exp \left[{\frac {y-\mu }{\sigma }}\right]\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>α</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>σ</mi> </mfrac> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>y</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =1-\exp \left[-\left({\frac {x+1}{\alpha }}\right)^{\beta }\right]=1-\exp \left[-\left({\frac {e^{y}}{e^{\mu }}}\right)^{\frac {1}{\sigma }}\right]=1-\exp \left[-\exp \left[{\frac {y-\mu }{\sigma }}\right]\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25ccaa7e3641fe8d829b42de25c253a07acfec32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:76.795ex; height:8.176ex;" alt="{\displaystyle =1-\exp \left[-\left({\frac {x+1}{\alpha }}\right)^{\beta }\right]=1-\exp \left[-\left({\frac {e^{y}}{e^{\mu }}}\right)^{\frac {1}{\sigma }}\right]=1-\exp \left[-\exp \left[{\frac {y-\mu }{\sigma }}\right]\right]}"></span></dd></dl> <p>which in estimation settings makes a lot of sense. This opens up the possibility of regression with frameworks developed for Weibull regression and extreme-value-theory.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Comparison_to_Other_Discrete_Distributions">Comparison to Other Discrete Distributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_Weibull_distribution&action=edit&section=3" title="Edit section: Comparison to Other Discrete Distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The discrete Weibull distribution can be compared with other common discrete distributions such as the Poisson, geometric, and negative binomial distributions, each of which has unique characteristics and applications. </p> <div class="mw-heading mw-heading3"><h3 id="Discrete_Weibull_vs._Poisson_Distribution">Discrete Weibull vs. Poisson Distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_Weibull_distribution&action=edit&section=4" title="Edit section: Discrete Weibull vs. Poisson Distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Poisson distribution is often used to model the number of rare event occurrences during a fixed period of time. It is characterized by a single parameter, λ, which is both the mean and variance of the distribution. The discrete Weibull distribution, on the other hand, is more flexible and can handle both over- and under-dispersion in count data. It has two parameters, q and β, which influence the shape and scale of the distribution. Unlike the Poisson distribution, which assumes events occur independently, the discrete Weibull can adapt to different event occurrence patterns. </p> <div class="mw-heading mw-heading3"><h3 id="Discrete_Weibull_vs._Geometric_Distribution">Discrete Weibull vs. Geometric Distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_Weibull_distribution&action=edit&section=5" title="Edit section: Discrete Weibull vs. Geometric Distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The geometric distribution models the probability of the first success in a sequence of Bernoulli trials and is characterized by a single parameter, p, which is the probability of success on an individual trial. In contrast, the discrete Weibull distribution can model a broader range of data patterns due to its two parameters. While the geometric distribution is specifically for modeling the number of trials until the first success, the discrete Weibull can be used in a wider variety of scenarios, including those where the probability of success changes over trials. </p> <div class="mw-heading mw-heading3"><h3 id="Discrete_Weibull_vs._Negative_Binomial_Distribution">Discrete Weibull vs. Negative Binomial Distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_Weibull_distribution&action=edit&section=6" title="Edit section: Discrete Weibull vs. Negative Binomial Distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The negative binomial distribution is used to model the number of Bernoulli trials needed before a particular number of successes is achieved. It is characterized by the probability of success and the number of successes. The discrete Weibull distribution, with its flexibility in modeling different data patterns, can be a better fit for data that does not conform to the specific scenario modeled by the negative binomial distribution. </p><p>Overall the discrete Weibull distribution is preferred over these alternatives when dealing with data that exhibit variability in dispersion (over- or under-dispersion) or when the data patterns do not fit the specific scenarios that Poisson, geometric, or negative binomial distributions are best suited for. Its adaptability in terms of shape and scale makes it a versatile tool in statistical modeling of discrete data.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_Weibull_distribution&action=edit&section=7" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Discrete Weibull distribution finds diverse applications in statistical analysis, as evidenced by various scholarly papers. One such paper illustrates the distribution's utility in modeling count data, specifically in the context of fertility plans. This study highlights how the Discrete Weibull distribution effectively captures complex relationships influenced by factors like education and family background. Unlike the Poisson distribution, it adeptly manages both overdispersed and underdispersed data, demonstrating its flexibility and efficacy in social science research. This application marks a significant extension of the distribution's usage beyond its traditional role in reliability engineering.<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><p>Further expanding its scope, "On Bivariate Discrete Weibull Distribution" explores the application of the Discrete Weibull distribution to bivariate data. The paper delves into sophisticated statistical techniques, including maximum likelihood estimation and Bayesian inference, for analyzing bivariate discrete data. This exploration underscores the distribution's compatibility with complex statistical methods. Moreover, the paper presents practical analysis scenarios, such as examining football match scores and nasal drainage severity, highlighting the distribution's broad applicability across varied fields. These instances underscore the distribution's practicality in real-world situations, moving beyond mere theoretical constructs.<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> </p><p>Another significant advancement is presented in "The Exponentiated Discrete Weibull Distribution," which introduces an enhanced version of the distribution, termed the Exponentiated Discrete Weibull Distribution (EDW). This generalization increases the model's flexibility, enabling it to represent a broader spectrum of data patterns, including various hazard rate functions like increasing, decreasing, bathtub-shaped, and inverted bathtub-shaped. The EDW distribution's ability to model both overdispersed and underdispersed data, relative to a Poisson distribution, broadens its applicability. It proves to be a versatile tool for various fields, including reliability engineering and failure time studies, further broadening the distribution's practical utility.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_Weibull_distribution&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull distribution</a></li> <li><a href="/wiki/Geometric_distribution" title="Geometric distribution">Geometric distribution</a></li> <li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution">q-Weibull distribution</a></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=Discrete_Weibull_distribution&action=edit&section=9" 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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></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="CITEREFKreer,_MarkusKizilersu,_AyseThomas,_Anthony_W.2024" class="citation journal cs1">Kreer, Markus; Kizilersu, Ayse; Thomas, Anthony W. 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