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It is also known, especially among [[physicist]]s, as the '''Lorentz distribution''' (after [[Hendrik Lorentz]]), '''Cauchy–Lorentz distribution''', '''Lorentz(ian) function''', or '''Breit–Wigner distribution'''. The Cauchy distribution <math>f(x; x_0,\gamma)</math> is the distribution of the {{mvar|x}}-intercept of a ray issuing from <math>(x_0,\gamma)</math> with a uniformly distributed angle. It is also the distribution of the [[Ratio distribution|ratio]] of two independent [[Normal distribution|normally distributed]] random variables with mean zero. The Cauchy distribution is often used in statistics as the canonical example of a "[[pathological (mathematics)|pathological]]" distribution since both its [[expected value]] and its [[variance]] are undefined (but see {{slink||Moments}} below). The Cauchy distribution does not have finite [[moment (mathematics)|moment]]s of order greater than or equal to one; only fractional absolute moments exist.<ref name=jkb1>{{cite book|author1=N. L. Johnson |author2=S. Kotz |author3=N. Balakrishnan |title=Continuous Univariate Distributions, Volume 1|publisher=Wiley|location=New York|year=1994}}, Chapter 16.</ref> The Cauchy distribution has no [[moment generating function]]. In [[mathematics]], it is closely related to the [[Poisson kernel]], which is the [[fundamental solution]] for the [[Laplace equation]] in the [[upper half-plane]]. It is one of the few [[stable distribution]]s with a probability density function that can be expressed analytically, the others being the [[normal distribution]] and the [[Lévy distribution]]. ==History== [[File:Mean estimator consistency.gif|thumb|upright=1.35|right|Estimating the mean and standard deviation through a sample from a Cauchy distribution (bottom) does not converge as the size of the sample grows, as in the [[normal distribution]] (top). There can be arbitrarily large jumps in the estimates, as seen in the graphs on the bottom. (Click to expand)]] A function with the form of the density function of the Cauchy distribution was studied geometrically by [[Pierre de Fermat|Fermat]] in 1659, and later was known as the [[witch of Agnesi]], after [[Maria Gaetana Agnesi]] included it as an example in her 1748 calculus textbook. Despite its name, the first explicit analysis of the properties of the Cauchy distribution was published by the French mathematician [[Siméon Denis Poisson|Poisson]] in 1824, with Cauchy only becoming associated with it during an academic controversy in 1853.<ref>Cauchy and the Witch of Agnesi in ''Statistics on the Table'', S M Stigler Harvard 1999 Chapter 18</ref> Poisson noted that if the mean of observations following such a distribution were taken, the [[standard deviation]] did not converge to any finite number. As such, [[Pierre-Simon Laplace|Laplace]]'s use of the [[central limit theorem]] with such a distribution was inappropriate, as it assumed a finite mean and variance. Despite this, Poisson did not regard the issue as important, in contrast to [[Irénée-Jules Bienaymé|Bienaymé]], who was to engage Cauchy in a long dispute over the matter. ==Constructions== Here are the most important constructions. === Rotational symmetry === If one stands in front of a line and kicks a ball with a direction (more precisely, an angle) uniformly at random towards the line, then the distribution of the point where the ball hits the line is a Cauchy distribution. More formally, consider a point at <math>(x_0, \gamma)</math> in the x-y plane, and select a line passing the point, with its direction (angle with the <math>x</math>-axis) chosen uniformly (between -90° and +90°) at random. The intersection of the line with the x-axis is the Cauchy distribution with location <math>x_0</math> and scale <math>\gamma</math>. This definition gives a simple way to sample from the standard Cauchy distribution. Let <math> u </math> be a sample from a uniform distribution from <math>[0,1]</math>, then we can generate a sample, <math>x</math> from the standard Cauchy distribution using :<math> x = \tan\left(\pi(u-\frac{1}{2})\right) </math> When <math>U</math> and <math>V</math> are two independent [[normal distribution|normally distributed]] [[random variable]]s with [[expected value]] 0 and [[variance]] 1, then the ratio <math>U/V</math> has the standard Cauchy distribution. More generally, if <math>(U, V)</math> is a rotationally symmetric distribution on the plane, then the ratio <math>U/V</math> has the standard Cauchy distribution. ===Probability density function (PDF)=== The Cauchy distribution is the probability distribution with the following [[probability density function]] (PDF)<ref name=jkb1/><ref name=feller>{{cite book|last=Feller|first=William|title=An Introduction to Probability Theory and Its Applications, Volume II|edition=2|publisher=John Wiley & Sons Inc.|location=New York|year=1971|pages=[https://archive.org/details/introductiontopr00fell/page/704 704]|isbn=978-0-471-25709-7|url-access=registration|url=https://archive.org/details/introductiontopr00fell/page/704}}</ref> :<math>f(x; x_0,\gamma) = \frac{1}{\pi\gamma \left[1 + \left(\frac{x - x_0}{\gamma}\right)^2\right]} = { 1 \over \pi } \left[ { \gamma \over (x - x_0)^2 + \gamma^2 } \right], </math> where <math>x_0</math> is the [[location parameter]], specifying the location of the peak of the distribution, and <math>\gamma</math> is the [[scale parameter]] which specifies the half-width at half-maximum (HWHM), alternatively <math>2\gamma</math> is [[full width at half maximum]] (FWHM). <math>\gamma</math> is also equal to half the [[interquartile range]] and is sometimes called the [[probable error]]. This function is also known as a [[Lorentzian function]],<ref>{{cite web |title=Lorentzian Function |url=https://mathworld.wolfram.com/LorentzianFunction.html |website=MathWorld |publisher=Wolfram Research |accessdate=27 October 2024}}</ref> and an example of a [[nascent delta function]], and therefore approaches a [[Dirac delta function]] in the limit as <math>\gamma \to 0</math>. [[Augustin-Louis Cauchy]] exploited such a density function in 1827 with an [[infinitesimal]] scale parameter, defining this [[Dirac delta function]]. ==== Properties of PDF ==== The maximum value or amplitude of the Cauchy PDF is <math>\frac{1}{\pi \gamma}</math>, located at <math>x=x_0</math>. It is sometimes convenient to express the PDF in terms of the complex parameter <math>\psi= x_0 + i\gamma</math> :<math> f(x;\psi)=\frac{1}{\pi}\,\textrm{Im}\left(\frac{1}{x-\psi}\right)=\frac{1}{\pi}\,\textrm{Re}\left(\frac{-i}{x-\psi}\right) </math> The special case when <math>x_0 = 0</math> and <math>\gamma = 1</math> is called the '''standard Cauchy distribution''' with the probability density function<ref name="mathmethods">{{cite book|last1=Riley|first1=Ken F.|last2=Hobson|first2=Michael P.|last3=Bence|first3=Stephen J.|title=Mathematical Methods for Physics and Engineering|url=https://archive.org/details/mathematicalmeth00rile_192|url-access=limited|edition=3|publisher=Cambridge University Press|location=Cambridge, UK|year=2006|pages=[https://archive.org/details/mathematicalmeth00rile_192/page/n1362 1333]|isbn=978-0-511-16842-0}}</ref><ref name="primer">{{cite book|last1=Balakrishnan|first1=N.|last2=Nevrozov|first2=V. B.|title=A Primer on Statistical Distributions|edition=1|publisher=John Wiley & Sons Inc.|location=Hoboken, New Jersey|year=2003|pages=[https://archive.org/details/primeronstatisti0000bala/page/305 305]|isbn=0-471-42798-5|url=https://archive.org/details/primeronstatisti0000bala/page/305}}</ref> :<math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math> In physics, a three-parameter Lorentzian function is often used: :<math>f(x; x_0,\gamma,I) = \frac{I}{\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]} = I \left[ { \gamma^2 \over (x - x_0)^2 + \gamma^2 } \right], </math> where <math>I</math> is the height of the peak. The three-parameter Lorentzian function indicated is not, in general, a probability density function, since it does not integrate to 1, except in the special case where <math>I = \frac{1}{\pi\gamma}.\!</math> ===Cumulative distribution function (CDF)=== The Cauchy distribution is the probability distribution with the following [[cumulative distribution function]] (CDF): :<math>F(x; x_0,\gamma)=\frac{1}{\pi} \arctan\left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2}</math> and the [[quantile function]] (inverse [[cumulative distribution function|cdf]]) of the Cauchy distribution is :<math>Q(p; x_0,\gamma) = x_0 + \gamma\,\tan\left[\pi\left(p-\tfrac{1}{2}\right)\right].</math> It follows that the first and third quartiles are <math>(x_0 - \gamma, x_0 + \gamma)</math>, and hence the [[interquartile range]] is <math>2\gamma</math>. For the standard distribution, the cumulative distribution function simplifies to [[Inverse trigonometric functions|arctangent function]] <math>\arctan(x)</math>: :<math>F(x; 0,1)=\frac{1}{\pi} \arctan\left(x\right)+\frac{1}{2}</math> === Other constructions === The standard Cauchy distribution is the [[Student's t-distribution|Student's ''t''-distribution]] with one degree of freedom, and so it may be constructed by any method that constructs the Student's t-distribution.<ref>{{Cite journal |last=Li |first=Rui |last2=Nadarajah |first2=Saralees |date=2020-03-01 |title=A review of Student’s t distribution and its generalizations |url=https://link.springer.com/article/10.1007/s00181-018-1570-0 |journal=Empirical Economics |language=en |volume=58 |issue=3 |pages=1461–1490 |doi=10.1007/s00181-018-1570-0 |issn=1435-8921}}</ref> If <math>\Sigma</math> is a <math>p\times p</math> positive-semidefinite covariance matrix with strictly positive diagonal entries, then for [[Independent and identically distributed random variables|independent and identically distributed]] <math>X,Y\sim N(0,\Sigma)</math> and any random <math>p</math>-vector <math>w</math> independent of <math>X</math> and <math>Y</math> such that <math>w_1+\cdots+w_p=1</math> and <math>w_i\geq 0, i=1,\ldots,p,</math> (defining a [[categorical distribution]]) it holds that :<math>\sum_{j=1}^p w_j\frac{X_j}{Y_j}\sim\mathrm{Cauchy}(0,1).</math><ref name=":0">{{Cite journal |author1=Pillai N. |author2=Meng, X.L. |year=2016 |title=An unexpected encounter with Cauchy and Lévy |journal=[[The Annals of Statistics]] |volume=44 |issue=5 |pages=2089–2097 |arxiv=1505.01957 |doi=10.1214/15-AOS1407 |s2cid=31582370}}</ref> ==Properties== The Cauchy distribution is an example of a distribution which has no [[mean]], [[variance]] or higher [[moment (mathematics)|moments]] defined. Its [[mode (statistics)|mode]] and [[median]] are well defined and are both equal to <math>x_0</math>. The Cauchy distribution is an [[infinitely divisible probability distribution]]. It is also a strictly [[stability (probability)|stable]] distribution.<ref>{{cite book |author1=Campbell B. Read |author2=N. Balakrishnan |author3=Brani Vidakovic |author4=Samuel Kotz |year=2006 |title=Encyclopedia of Statistical Sciences |page=778 |edition=2nd |publisher=[[John Wiley & Sons]] |isbn=978-0-471-15044-2|title-link=Encyclopedia of Statistical Sciences }}</ref> Like all stable distributions, the [[location-scale family]] to which the Cauchy distribution belongs is closed under [[linear transformations]] with [[real number|real]] coefficients. In addition, the family of Cauchy-distributed random variables is closed under [[Möbius transformation|linear fractional transformations]] with real coefficients.<ref>{{cite journal|first1=Franck B. | last1=Knight|title=A characterization of the Cauchy type|journal=[[Proceedings of the American Mathematical Society]]|volume = 55|issue=1|year = 1976|pages= 130–135|doi=10.2307/2041858|jstor=2041858|doi-access=free}}</ref> In this connection, see also [[McCullagh's parametrization of the Cauchy distributions]]. === Sum of Cauchy-distributed random variables === If <math>X_1, X_2, \ldots, X_n</math> are an [[Independent and identically distributed random variables|IID]] sample from the standard Cauchy distribution, then their [[sample mean]] <math>\bar X = \frac 1n \sum_i X_i</math> is also standard Cauchy distributed. In particular, the average does not converge to the mean, and so the standard Cauchy distribution does not follow the law of large numbers. This can be proved by repeated integration with the PDF, or more conveniently, by using the [[Characteristic function (probability theory)|characteristic function]] of the standard Cauchy distribution (see below):<math display="block">\varphi_X(t) = \operatorname{E}\left[e^{iXt} \right ] = e^{-|t|}.</math>With this, we have <math>\varphi_{\sum_i X_i}(t) = e^{-n |t|} </math>, and so <math>\bar X</math> has a standard Cauchy distribution. More generally, if <math>X_1, X_2, \ldots, X_n</math> are independent and Cauchy distributed with location parameters <math>x_1, \ldots, x_n</math> and scales <math>\gamma_1, \ldots, \gamma_n</math>, and <math>a_1, \ldots, a_n</math> are real numbers, then <math>\sum_i a_iX_i</math> is Cauchy distributed with location <math>\sum_i a_ix_i</math> and scale<math>\sum_i |a_i|\gamma_i</math>. We see that there is no law of large numbers for any weighted sum of independent Cauchy distributions. This shows that the condition of finite variance in the [[central limit theorem]] cannot be dropped. It is also an example of a more generalized version of the central limit theorem that is characteristic of all [[stable distribution]]s, of which the Cauchy distribution is a special case. === Central limit theorem === If <math>X_1, X_2, \ldots </math> are and IID sample with PDF <math>\rho</math> such that <math>\lim_{c \to \infty}\frac{1}{c} \int_{-c}^c x^2\rho(x) \, dx = \frac{2\gamma}{\pi} </math> is finite, but nonzero, then <math>\frac 1n \sum_{i=1}^n X_i</math> converges in distribution to a Cauchy distribution with scale <math>\gamma</math>.<ref>{{cite web | title=Updates to the Cauchy Central Limit | website=Quantum Calculus | date=13 November 2022 | url=https://www.quantumcalculus.org/updates-to-the-cauchy-central-limit/ | access-date=21 June 2023}}</ref> ===Characteristic function=== Let <math>X</math> denote a Cauchy distributed random variable. The [[Characteristic function (probability theory)|characteristic function]] of the Cauchy distribution is given by :<math>\varphi_X(t) = \operatorname{E}\left[e^{iXt} \right ] =\int_{-\infty}^\infty f(x;x_0,\gamma)e^{ixt}\,dx = e^{ix_0t - \gamma |t|}.</math> which is just the [[Fourier transform]] of the probability density. The original probability density may be expressed in terms of the characteristic function, essentially by using the inverse Fourier transform: :<math>f(x; x_0,\gamma) = \frac{1}{2\pi}\int_{-\infty}^\infty \varphi_X(t;x_0,\gamma)e^{-ixt} \, dt \!</math> The ''n''th moment of a distribution is the ''n''th derivative of the characteristic function evaluated at <math>t=0</math>. Observe that the characteristic function is not [[Differentiable function|differentiable]] at the origin: this corresponds to the fact that the Cauchy distribution does not have well-defined moments higher than the zeroth moment. === Kullback–Leibler divergence === The [[Kullback–Leibler divergence]] between two Cauchy distributions has the following symmetric closed-form formula:<ref>{{cite arXiv |last1=Frederic |first1=Chyzak |last2=Nielsen |first2=Frank |year=2019 |title=A closed-form formula for the Kullback-Leibler divergence between Cauchy distributions |class=cs.IT |eprint=1905.10965}}</ref> :<math> \mathrm{KL}\left(p_{x_{0,1}, \gamma_{1}}: p_{x_{0,2}, \gamma_{2}}\right)=\log \frac{\left(\gamma_{1}+\gamma_{2}\right)^{2}+\left(x_{0,1}-x_{0,2}\right)^{2}}{4 \gamma_{1} \gamma_{2}}. </math> Any [[f-divergence]] between two Cauchy distributions is symmetric and can be expressed as a function of the chi-squared divergence.<ref>{{cite journal |last1=Nielsen |first1=Frank |last2=Okamura |first2=Kazuki |year=2023 |title=On f-Divergences Between Cauchy Distributions |journal=IEEE Transactions on Information Theory |volume=69 |issue=5 |pages=3150–3171 |doi=10.1109/TIT.2022.3231645 |arxiv=2101.12459|s2cid=231728407 }}</ref> Closed-form expression for the [[total variation]], [[Jensen–Shannon divergence]], [[Hellinger distance]], etc. are available. === Entropy === The entropy of the Cauchy distribution is given by: : <math> \begin{align} H(\gamma) & =-\int_{-\infty}^\infty f(x;x_0,\gamma) \log(f(x;x_0,\gamma)) \, dx \\[6pt] & =\log(4\pi\gamma) \end{align} </math> The derivative of the [[quantile function]], the quantile density function, for the Cauchy distribution is: :<math>Q'(p; \gamma) = \gamma\,\pi\,{\sec}^2\left[\pi\left(p-\tfrac 1 2 \right)\right].\!</math> The [[differential entropy]] of a distribution can be defined in terms of its quantile density,<ref>{{cite journal |last1=Vasicek |first1=Oldrich |year=1976 |title=A Test for Normality Based on Sample Entropy |journal=Journal of the Royal Statistical Society, Series B |volume=38 |issue=1 |pages=54–59|doi=10.1111/j.2517-6161.1976.tb01566.x }}</ref> specifically: :<math>H(\gamma) = \int_0^1 \log\,(Q'(p; \gamma))\,\mathrm dp = \log(4\pi\gamma)</math> The Cauchy distribution is the [[maximum entropy probability distribution]] for a random variate <math>X</math> for which<ref>{{cite journal |last1=Park |first1=Sung Y. |last2=Bera |first2=Anil K. |year=2009 |title=Maximum entropy autoregressive conditional heteroskedasticity model |url=http://www.econ.yorku.ca/cesg/papers/berapark.pdf |url-status=dead |journal=Journal of Econometrics |publisher=Elsevier |volume=150 |issue=2 |pages=219–230 |doi=10.1016/j.jeconom.2008.12.014 |archive-url=https://web.archive.org/web/20110930062639/http://www.econ.yorku.ca/cesg/papers/berapark.pdf |archive-date=2011-09-30 |access-date=2011-06-02}}</ref> :<math>\operatorname{E}[\log(1+(X-x_0)^2/\gamma^2)]=\log 4</math> ==Moments== The Cauchy distribution is usually used as an illustrative counterexample in elementary probability courses, as a distribution with no well-defined (or "indefinite") moments. === Sample moments === If we take an IID sample <math>X_1, X_2, \ldots </math> from the standard Cauchy distribution, then the sequence of their sample mean is <math>S_n = \frac 1n \sum_{i=1}^n X_i</math>, which also has the standard Cauchy distribution. Consequently, no matter how many terms we take, the sample average does not converge. Similarly, the sample variance <math>V_n = \frac 1n \sum_{i=1}^n (X_i - S_n)^2</math> also does not converge. [[File:Sample mean and variance of IID samples from a standard Cauchy distribution..png|thumb|upright=2.14|A typical trajectory of sample means looks like long periods of slow convergence to zero, punctuated by large jumps away from zero, but never getting too far away. A typical trajectory of sample variances looks similar, but the jumps accumulate faster than the decay, diverging to infinity.]] A typical trajectory of <math>S_1, S_2, ...</math> looks like long periods of slow convergence to zero, punctuated by large jumps away from zero, but never getting too far away. A typical trajectory of <math>V_1, V_2, ...</math> looks similar, but the jumps accumulate faster than the decay, diverging to infinity. These two kinds of trajectories are plotted in the figure. Moments of sample lower than order 1 would converge to zero. Moments of sample higher than order 2 would diverge to infinity even faster than sample variance. ===Mean=== If a [[probability distribution]] has a [[probability density function|density function]] <math>f(x)</math>, then the mean, if it exists, is given by {{NumBlk||<math display="block">\int_{-\infty}^\infty x f(x)\,dx. </math>|{{EquationRef|1}}}} We may evaluate this two-sided [[improper integral]] by computing the sum of two one-sided improper integrals. That is, {{NumBlk||<math display="block">\int_{-\infty}^a x f(x)\,dx +\int_a^\infty x f(x) \, dx </math>|{{EquationRef|2}}}} for an arbitrary real number <math>a</math>. For the integral to exist (even as an infinite value), at least one of the terms in this sum should be finite, or both should be infinite and have the same sign. But in the case of the Cauchy distribution, both the terms in this sum ({{EquationNote|2}}) are infinite and have opposite sign. Hence ({{EquationNote|1}}) is undefined, and thus so is the mean.<ref name="uah">{{cite web | url=http://www.randomservices.org/random/special/Cauchy.html | title=Cauchy Distribution | author=Kyle Siegrist | work=Random | access-date=5 July 2021 | archive-date=9 July 2021 | archive-url=https://web.archive.org/web/20210709183100/http://www.randomservices.org/random/special/Cauchy.html | url-status=live }}</ref> When the mean of a probability distribution function (PDF) is undefined, no one can compute a reliable average over the experimental data points, regardless of the sample's size. Note that the [[Cauchy principal value]] of the mean of the Cauchy distribution is <math display="block">\lim_{a\to\infty}\int_{-a}^a x f(x)\,dx </math> which is zero. On the other hand, the related integral <math display="block">\lim_{a\to\infty}\int_{-2a}^a x f(x)\,dx </math> is ''not'' zero, as can be seen by computing the integral. This again shows that the mean ({{EquationNote|1}}) cannot exist. Various results in probability theory about [[expected value]]s, such as the [[strong law of large numbers]], fail to hold for the Cauchy distribution.<ref name="uah"/> ===Smaller moments=== The absolute moments for <math>p\in(-1,1)</math> are defined. For <math>X\sim\mathrm{Cauchy}(0,\gamma)</math> we have :<math>\operatorname{E}[|X|^p] = \gamma^p \mathrm{sec}(\pi p/2).</math> ===Higher moments=== The Cauchy distribution does not have finite moments of any order. Some of the higher [[raw moment]]s do exist and have a value of infinity, for example, the raw second moment: : <math> \begin{align} \operatorname{E}[X^2] & \propto \int_{-\infty}^\infty \frac{x^2}{1+x^2}\,dx = \int_{-\infty}^\infty 1 - \frac{1}{1+x^2}\,dx \\[8pt] & = \int_{-\infty}^\infty dx - \int_{-\infty}^\infty \frac{1}{1+x^2}\,dx = \int_{-\infty}^\infty dx-\pi = \infty. \end{align} </math> By re-arranging the formula, one can see that the second moment is essentially the infinite integral of a constant (here 1). Higher even-powered raw moments will also evaluate to infinity. Odd-powered raw moments, however, are undefined, which is distinctly different from existing with the value of infinity. The odd-powered raw moments are undefined because their values are essentially equivalent to <math>\infty - \infty</math> since the two halves of the integral both diverge and have opposite signs. The first raw moment is the mean, which, being odd, does not exist. (See also the discussion above about this.) This in turn means that all of the [[central moment]]s and [[standardized moment]]s are undefined since they are all based on the mean. The variance—which is the second central moment—is likewise non-existent (despite the fact that the raw second moment exists with the value infinity). The results for higher moments follow from [[Hölder's inequality]], which implies that higher moments (or halves of moments) diverge if lower ones do. ===Moments of truncated distributions=== Consider the [[truncated distribution]] defined by restricting the standard Cauchy distribution to the interval {{math|[−10<sup>100</sup>, 10<sup>100</sup>]}}. Such a truncated distribution has all moments (and the central limit theorem applies for [[i.i.d.]] observations from it); yet for almost all practical purposes it behaves like a Cauchy distribution.<ref>{{citation | last= Hampel | first= Frank | title= Is statistics too difficult? | journal= Canadian Journal of Statistics | year= 1998 | volume= 26 | issue= 3 | pages= 497–513 | doi= 10.2307/3315772 | jstor= 3315772 | hdl= 20.500.11850/145503 | s2cid= 53117661 | url= https://www.research-collection.ethz.ch/bitstream/20.500.11850/145503/1/eth-24416-01.pdf | hdl-access= free | access-date= 2019-09-25 | archive-date= 2022-01-25 | archive-url= https://web.archive.org/web/20220125125836/https://www.research-collection.ethz.ch/bitstream/handle/20.500.11850/145503/eth-24416-01.pdf;jsessionid=90EA750F49FB4DCEE3C23A4F8B49916B?sequence=1 | url-status= live }}.</ref> ==Estimation of parameters == Because the parameters of the Cauchy distribution do not correspond to a mean and variance, attempting to estimate the parameters of the Cauchy distribution by using a sample mean and a sample variance will not succeed.<ref>{{cite web| url = http://www.statistics4u.info/fundstat_eng/ee_distri_cauchy.html| title = Illustration of instability of sample means| access-date = 2014-11-22| archive-date = 2017-03-24| archive-url = https://web.archive.org/web/20170324193842/http://www.statistics4u.info/fundstat_eng/ee_distri_cauchy.html| url-status = live}}</ref> For example, if an i.i.d. sample of size ''n'' is taken from a Cauchy distribution, one may calculate the sample mean as: :<math>\bar{x}=\frac 1 n \sum_{i=1}^n x_i</math> Although the sample values <math>x_i</math> will be concentrated about the central value <math>x_0</math>, the sample mean will become increasingly variable as more observations are taken, because of the increased probability of encountering sample points with a large absolute value. In fact, the distribution of the sample mean will be equal to the distribution of the observations themselves; i.e., the sample mean of a large sample is no better (or worse) an estimator of <math>x_0</math> than any single observation from the sample. Similarly, calculating the sample variance will result in values that grow larger as more observations are taken. Therefore, more robust means of estimating the central value <math>x_0</math> and the scaling parameter <math>\gamma</math> are needed. One simple method is to take the median value of the sample as an estimator of <math>x_0</math> and half the sample [[interquartile range]] as an estimator of <math>\gamma</math>. Other, more precise and robust methods have been developed <ref>{{cite journal |last1=Cane |first1=Gwenda J. |year=1974 |title=Linear Estimation of Parameters of the Cauchy Distribution Based on Sample Quantiles |journal=Journal of the American Statistical Association |volume=69 |issue=345 |pages= 243–245 |jstor=2285535 |doi=10.1080/01621459.1974.10480163}}</ref><ref>{{cite journal |last=Zhang |first=Jin |year=2010 |title=A Highly Efficient L-estimator for the Location Parameter of the Cauchy Distribution |journal=Computational Statistics |volume=25 |issue=1 |pages=97–105 |doi=10.1007/s00180-009-0163-y|s2cid=123586208 }}</ref> For example, the [[truncated mean]] of the middle 24% of the sample [[order statistics]] produces an estimate for <math>x_0</math> that is more efficient than using either the sample median or the full sample mean.<ref name=rothenberg>{{cite journal|last1=Rothenberg |first1=Thomas J. |last2=Fisher|first2=Franklin, M.|last3=Tilanus|first3=C.B.|year=1964|volume=59|issue=306|journal=Journal of the American Statistical Association|title=A note on estimation from a Cauchy sample|pages=460–463|doi=10.1080/01621459.1964.10482170}}</ref><ref name=bloch>{{cite journal|last1=Bloch|first1=Daniel|year=1966|volume=61 |issue=316 |journal=Journal of the American Statistical Association|title=A note on the estimation of the location parameters of the Cauchy distribution|pages=852&ndash;855|jstor=2282794|doi=10.1080/01621459.1966.10480912}}</ref> However, because of the [[fat tails]] of the Cauchy distribution, the efficiency of the estimator decreases if more than 24% of the sample is used.<ref name=rothenberg/><ref name=bloch/> [[Maximum likelihood]] can also be used to estimate the parameters <math>x_0</math> and <math>\gamma</math>. However, this tends to be complicated by the fact that this requires finding the roots of a high degree polynomial, and there can be multiple roots that represent local maxima.<ref name=ferguson>{{cite journal|last1=Ferguson|first1=Thomas S.|author-link= Thomas S. Ferguson |year=1978 |journal=Journal of the American Statistical Association |volume=73|issue=361|pages=211–213 |title=Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4|jstor=2286549 |doi=10.1080/01621459.1978.10480031}}</ref> Also, while the maximum likelihood estimator is asymptotically efficient, it is relatively inefficient for small samples.<ref>{{cite journal|title=The Pitman estimator of the Cauchy location parameter|last1=Cohen Freue|first1=Gabriella V.|journal=Journal of Statistical Planning and Inference|volume=137|issue=6|year=2007|page=1901|url=http://faculty.ksu.edu.sa/69424/USEPAP/Coushy%20dist.pdf|doi=10.1016/j.jspi.2006.05.002|url-status=dead|archive-url=https://web.archive.org/web/20110816002255/http://faculty.ksu.edu.sa/69424/USEPAP/Coushy%20dist.pdf|archive-date=2011-08-16}}</ref><ref>{{cite book|title=Introduction to Robust Estimation & Hypothesis Testing |last1=Wilcox |first1=Rand |year=2012 |publisher=Elsevier}}</ref> The log-likelihood function for the Cauchy distribution for sample size <math>n</math> is: :<math>\hat\ell(x_1,\dotsc,x_n \mid \!x_0,\gamma ) = - n \log (\gamma \pi) - \sum_{i=1}^n \log \left(1 + \left(\frac{x_i - x_0}{\gamma}\right)^2\right)</math> Maximizing the log likelihood function with respect to <math>x_0</math> and <math>\gamma</math> by taking the first derivative produces the following system of equations: :<math> \frac{d \ell}{d x_{0}} = \sum_{i=1}^n \frac{2(x_i - x_0)}{\gamma^2 + \left(x_i - \!x_0\right)^2} =0</math> :<math> \frac{d \ell}{d \gamma} = \sum_{i=1}^n \frac{2\left(x_i - x_0\right)^2}{\gamma (\gamma^2 + \left(x_i - x_0\right)^2)} - \frac{n}{\gamma} = 0</math> Note that :<math> \sum_{i=1}^n \frac{\left(x_i - x_0\right)^2}{\gamma^2 + \left(x_i - x_0\right)^2} </math> is a monotone function in <math>\gamma</math> and that the solution <math>\gamma</math> must satisfy :<math> \min |x_i-x_0|\le \gamma\le \max |x_i-x_0|. </math> Solving just for <math>x_0</math> requires solving a polynomial of degree <math>2n-1</math>,<ref name=ferguson/> and solving just for <math>\,\!\gamma</math> requires solving a polynomial of degree <math>2n</math>. Therefore, whether solving for one parameter or for both parameters simultaneously, a [[numerical analysis|numerical]] solution on a computer is typically required. The benefit of maximum likelihood estimation is asymptotic efficiency; estimating <math>x_0</math> using the sample median is only about 81% as asymptotically efficient as estimating <math>x_0</math> by maximum likelihood.<ref name=bloch/><ref>{{cite journal|last1=Barnett|first1=V. D.|year=1966|journal=Journal of the American Statistical Association |volume=61|issue=316|pages=1205–1218|title=Order Statistics Estimators of the Location of the Cauchy Distribution|jstor=2283210|doi=10.1080/01621459.1966.10482205}}</ref> The truncated sample mean using the middle 24% order statistics is about 88% as asymptotically efficient an estimator of <math>x_0</math> as the maximum likelihood estimate.<ref name=bloch/> When [[Newton's method]] is used to find the solution for the maximum likelihood estimate, the middle 24% order statistics can be used as an initial solution for <math>x_0</math>. The shape can be estimated using the median of absolute values, since for location 0 Cauchy variables <math>X\sim\mathrm{Cauchy}(0,\gamma)</math>, the <math>\operatorname{median}(|X|) = \gamma</math> the shape parameter. ==Multivariate Cauchy distribution== A [[random vector]] <math>X=(X_1, \ldots, X_k)^T</math> is said to have the multivariate Cauchy distribution if every linear combination of its components <math>Y=a_1X_1+ \cdots + a_kX_k</math> has a Cauchy distribution. That is, for any constant vector <math>a\in \mathbb R^k</math>, the random variable <math>Y=a^TX</math> should have a univariate Cauchy distribution.<ref name=ferg2>{{cite journal|last1=Ferguson|first1=Thomas S.|title=A Representation of the Symmetric Bivariate Cauchy Distribution|journal=The Annals of Mathematical Statistics |volume= 33|issue= 4|pages=1256–1266|year=1962 |jstor=2237984|doi=10.1214/aoms/1177704357|url=http://projecteuclid.org/download/pdf_1/euclid.aoms/1177704357|access-date=2017-01-07 |doi-access=free}}</ref> The characteristic function of a multivariate Cauchy distribution is given by: :<math>\varphi_X(t) = e^{ix_0(t)-\gamma(t)}, \!</math> where <math>x_0(t)</math> and <math>\gamma(t)</math> are real functions with <math>x_0(t)</math> a [[homogeneous function]] of degree one and <math>\gamma(t)</math> a positive homogeneous function of degree one.<ref name=ferg2/> More formally:<ref name=ferg2/> :<math>x_0(at) = ax_0(t),</math> :<math>\gamma (at) = |a|\gamma (t),</math> for all <math>t</math>. An example of a bivariate Cauchy distribution can be given by:<ref name=bivar>{{cite journal|title=Non-linear Integral Equations to Approximate Bivariate Densities with Given Marginals and Dependence Function|last1=Molenberghs|first1=Geert|last2=Lesaffre|first2=Emmanuel|journal=Statistica Sinica|volume=7|year=1997|pages=713&ndash;738|url=http://www3.stat.sinica.edu.tw/statistica/oldpdf/A7n310.pdf|url-status=dead|archive-url=https://web.archive.org/web/20090914055538/http://www3.stat.sinica.edu.tw/statistica/oldpdf/A7n310.pdf|archive-date=2009-09-14}}</ref> :<math>f(x, y; x_0,y_0,\gamma)= { 1 \over 2 \pi } \left[ { \gamma \over ((x - x_0)^2 + (y - y_0)^2 +\gamma^2)^{3/2} } \right] .</math> Note that in this example, even though the covariance between <math>x</math> and <math>y</math> is 0, <math>x</math> and <math>y</math> are not [[Independence (probability theory)|statistically independent]].<ref name=bivar/> We also can write this formula for complex variable. Then the probability density function of complex cauchy is : :<math>f(z; z_0,\gamma)= { 1 \over 2 \pi } \left[ { \gamma \over (|z-z_0|^2 +\gamma^2)^{3/2} } \right] .</math> Like how the standard Cauchy distribution is the Student t-distribution with one degree of freedom, the multidimensional Cauchy density is the [[multivariate Student distribution]] with one degree of freedom. The density of a <math>k</math> dimension Student distribution with one degree of freedom is: :<math>f({\mathbf x}; {\mathbf\mu},{\mathbf\Sigma}, k)= \frac{\Gamma\left(\frac{1+k}{2}\right)}{\Gamma(\frac{1}{2})\pi^{\frac{k}{2}}\left|{\mathbf\Sigma}\right|^{\frac{1}{2}}\left[1+({\mathbf x}-{\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x}-{\mathbf\mu})\right]^{\frac{1+k}{2}}} .</math> The properties of multidimensional Cauchy distribution are then special cases of the multivariate Student distribution. ==Transformation properties== *If <math>X \sim \operatorname{Cauchy}(x_0,\gamma)</math> then <math> kX + \ell \sim \textrm{Cauchy}(x_0 k+\ell, \gamma |k|)</math><ref>{{Citation | last1 = Lemons | first1 = Don S. | title = An Introduction to Stochastic Processes in Physics | journal = American Journal of Physics | publisher = The Johns Hopkins University Press | year = 2002 | volume = 71 | issue = 2 | isbn = 0-8018-6866-1 | page=35 | doi = 10.1119/1.1526134 | bibcode = 2003AmJPh..71..191L }}</ref> *If <math>X \sim \operatorname{Cauchy}(x_0, \gamma_0)</math> and <math>Y \sim \operatorname{Cauchy}(x_1,\gamma_1)</math> are independent, then <math> X+Y \sim \operatorname{Cauchy}(x_0+x_1,\gamma_0 +\gamma_1)</math> and <math> X-Y \sim \operatorname{Cauchy}(x_0-x_1, \gamma_0+\gamma_1)</math> *If <math>X \sim \operatorname{Cauchy}(0,\gamma)</math> then <math> \tfrac{1}{X} \sim \operatorname{Cauchy}(0, \tfrac{1}{\gamma})</math> *[[McCullagh's parametrization of the Cauchy distributions]]:<ref name="McCullagh1992">[[Peter McCullagh|McCullagh, P.]], [https://archive.today/20120707071014/http://biomet.oxfordjournals.org/cgi/content/abstract/79/2/247 "Conditional inference and Cauchy models"], ''[[Biometrika]]'', volume 79 (1992), pages 247&ndash;259. [http://www.stat.uchicago.edu/~pmcc/pubs/paper18.pdf PDF] {{Webarchive |url=https://web.archive.org/web/20100610000327/http://www.stat.uchicago.edu/~pmcc/pubs/paper18.pdf |date=2010-06-10 }} from McCullagh's homepage.</ref> Expressing a Cauchy distribution in terms of one complex parameter <math>\psi = x_0+i\gamma</math>, define <math>X \sim \operatorname{Cauchy}(\psi)</math> to mean <math>X \sim \operatorname{Cauchy}(x_0,|\gamma|)</math>. If <math>X \sim \operatorname{Cauchy}(\psi)</math> then: <math display="block">\frac{aX+b}{cX+d} \sim \operatorname{Cauchy}\left(\frac{a\psi+b}{c\psi+d}\right)</math> where <math>a</math>, <math>b</math>, <math>c</math> and <math>d</math> are real numbers. * Using the same convention as above, if <math>X \sim \operatorname{Cauchy}(\psi)</math> then:<ref name="McCullagh1992"/> <math display="block">\frac{X-i}{X+i} \sim \operatorname{CCauchy}\left(\frac{\psi-i}{\psi+i}\right)</math>where <math>\operatorname{CCauchy}</math> is the [[circular Cauchy distribution]]. == Lévy measure == The Cauchy distribution is the [[stable distribution]] of index 1. The [[Lévy process#L.C3.A9vy.E2.80.93Khintchine representation|Lévy–Khintchine representation]] of such a stable distribution of parameter <math> \gamma </math> is given, for <math> X \sim \operatorname{Stable}(\gamma, 0, 0)\,</math> by: : <math>\operatorname{E}\left( e^{ixX} \right) = \exp\left( \int_{ \mathbb{R} } (e^{ixy} - 1) \Pi_\gamma(dy) \right)</math> where :<math>\Pi_\gamma(dy) = \left( c_{1, \gamma} \frac{1}{y^{1 + \gamma}} 1_{ \left\{y > 0\right\} } + c_{2,\gamma} \frac{1}{|y|^{1 + \gamma}} 1_{\left\{ y < 0 \right\}} \right) \, dy </math> and <math> c_{1, \gamma}, c_{2, \gamma} </math> can be expressed explicitly.<ref>{{cite book |author=Kyprianou, Andreas |year=2009 |title=Lévy processes and continuous-state branching processes:part I |page=11 |url=http://www.maths.bath.ac.uk/~ak257/LCSB/part1.pdf |access-date=2016-05-04 |archive-date=2016-03-03 |archive-url=https://web.archive.org/web/20160303235654/http://www.maths.bath.ac.uk/~ak257/LCSB/part1.pdf |url-status=live }}</ref> In the case <math> \gamma = 1 </math> of the Cauchy distribution, one has <math> c_{1, \gamma} = c_{2, \gamma} </math>. This last representation is a consequence of the formula : <math>\pi |x| = \operatorname{PV }\int_{\mathbb{R} \smallsetminus\lbrace 0 \rbrace} (1 - e^{ixy}) \, \frac{dy}{y^2} </math> ==Related distributions== *<math>\operatorname{Cauchy}(0,1) \sim \textrm{t}(\mathrm{df}=1)\,</math> [[Student's t distribution|Student's ''t'' distribution]] *<math>\operatorname{Cauchy}(\mu,\sigma) \sim \textrm{t}_{(\mathrm{df}=1)}(\mu,\sigma)\,</math> [[Student's t distribution#location-scale|non-standardized Student's ''t'' distribution]] *If <math>X, Y \sim \textrm{N}(0,1)\, X, Y</math> independent, then <math> \tfrac X Y\sim \textrm{Cauchy}(0,1)\,</math> *If <math>X \sim \textrm{U}(0,1)\,</math> then <math> \tan \left( \pi \left(X-\tfrac{1}{2}\right) \right) \sim \textrm{Cauchy}(0,1)\,</math> *If <math>X \sim \operatorname{Log-Cauchy}(0, 1)</math> then <math>\ln(X) \sim \textrm{Cauchy}(0, 1)</math> *If <math>X \sim \operatorname{Cauchy}(x_0,\gamma)</math> then <math>\tfrac1X \sim \operatorname{Cauchy}\left(\tfrac{x_0}{x_0^2+\gamma^2},\tfrac{\gamma}{x_0^2+\gamma^2}\right)</math> *The Cauchy distribution is a limiting case of a [[Pearson distribution]] of type 4{{Citation needed|date=March 2011}} *The Cauchy distribution is a special case of a [[Pearson distribution]] of type 7.<ref name=jkb1/> *The Cauchy distribution is a [[stable distribution]]: if <math>X \sim \textrm{Stable}(1, 0, \gamma, \mu)</math>, then <math>X \sim \operatorname{Cauchy}(\mu, \gamma)</math>. *The Cauchy distribution is a singular limit of a [[hyperbolic distribution]]{{Citation needed|date=April 2011}} *The [[wrapped Cauchy distribution]], taking values on a circle, is derived from the Cauchy distribution by wrapping it around the circle. *If <math>X \sim \textrm{N}(0,1)</math>, <math>Z \sim \operatorname{Inverse-Gamma}(1/2, s^2/2)</math>, then <math>Y = \mu + X \sqrt Z \sim \operatorname{Cauchy}(\mu,s)</math>. For half-Cauchy distributions, the relation holds by setting <math>X \sim \textrm{N}(0,1) I\{X\ge0\}</math>. ==Relativistic Breit–Wigner distribution== {{Main article|Relativistic Breit–Wigner distribution}} In [[nuclear physics|nuclear]] and [[particle physics]], the energy profile of a [[resonance]] is described by the [[relativistic Breit–Wigner distribution]], while the Cauchy distribution is the (non-relativistic) Breit–Wigner distribution.{{Citation needed|date=March 2011}} ==Occurrence and applications== *In [[spectroscopy]], the Cauchy distribution describes the shape of [[spectral line]]s which are subject to [[homogeneous broadening]] in which all atoms interact in the same way with the frequency range contained in the line shape. Many mechanisms cause homogeneous broadening, most notably [[Line broadening#Pressure broadening|collision broadening]].<ref>{{cite book |author=E. Hecht |year=1987 |title=Optics |page=603 |edition=2nd |publisher=[[Addison-Wesley]] }}</ref> [[Spectral line#Natural broadening|Lifetime or natural broadening]] also gives rise to a line shape described by the Cauchy distribution. *Applications of the Cauchy distribution or its transformation can be found in fields working with [[exponential growth]]. A 1958 paper by White <ref>{{cite journal |author=White, J.S. |date=December 1958 |title=The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case |journal=The Annals of Mathematical Statistics |volume=29 |issue=4 |pages=1188–1197 |doi=10.1214/aoms/1177706450 |doi-access=free}}</ref> derived the test statistic for estimators of <math>\hat{\beta}</math> for the equation <math>x_{t+1}=\beta{x}_t+\varepsilon_{t+1},\beta>1</math> and where the maximum likelihood estimator is found using ordinary least squares showed the sampling distribution of the statistic is the Cauchy distribution. [[File:Cauchy distribution.png|thumb|upright=1.15|Fitted cumulative Cauchy distribution to maximum one-day rainfalls using [[CumFreq]], see also [[distribution fitting]]<ref name="cumfreq">{{cite web |title=CumFreq, free software for cumulative frequency analysis and probability distribution fitting |url=https://www.waterlog.info/cumfreq.htm |url-status=live |archive-url=https://web.archive.org/web/20180221100105/https://www.waterlog.info/cumfreq.htm|archive-date=2018-02-21}}</ref>]] *The Cauchy distribution is often the distribution of observations for objects that are spinning. The classic reference for this is called the Gull's lighthouse problem<ref>Gull, S.F. (1988) Bayesian Inductive Inference and Maximum Entropy. Kluwer Academic Publishers, Berlin. https://doi.org/10.1007/978-94-009-3049-0_4 {{Webarchive|url=https://web.archive.org/web/20220125125834/https://link.springer.com/chapter/10.1007%2F978-94-009-3049-0_4 |date=2022-01-25 }}</ref> and as in the above section as the Breit–Wigner distribution in particle physics. *In [[hydrology]] the Cauchy distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. The blue picture illustrates an example of fitting the Cauchy distribution to ranked monthly maximum one-day rainfalls showing also the 90% [[confidence belt]] based on the [[binomial distribution]]. The rainfall data are represented by [[plotting position]]s as part of the [[cumulative frequency analysis]]. *The expression for the imaginary part of complex [[Permittivity|electrical permittivity]], according to the Lorentz model, is a Cauchy distribution. *As an additional distribution to model [[fat tails]] in [[computational finance]], Cauchy distributions can be used to model VAR ([[value at risk]]) producing a much larger probability of extreme risk than [[Gaussian Distribution]].<ref>Tong Liu (2012), An intermediate distribution between Gaussian and Cauchy distributions. https://arxiv.org/pdf/1208.5109.pdf {{Webarchive|url=https://web.archive.org/web/20200624234315/https://arxiv.org/pdf/1208.5109.pdf |date=2020-06-24 }}</ref> ==See also== * [[Lévy flight]] and [[Lévy process]] * [[Laplace distribution]], the Fourier transform of the Cauchy distribution * [[Cauchy process]] * [[Stable process]] * [[Slash distribution]] ==References== {{Reflist|30em}} ==External links== * {{springer|title=Cauchy distribution|id=p/c020850}} * [http://jeff560.tripod.com/c.html Earliest Uses: The entry on Cauchy distribution has some historical information.] * {{MathWorld | urlname=CauchyDistribution | title=Cauchy Distribution}} * [https://www.gnu.org/software/gsl/manual/gsl-ref.html#SEC294 GNU Scientific Library &ndash; Reference Manual] * [http://www.jstatsoft.org/v16/i04/paper Ratios of Normal Variables by George Marsaglia] {{ProbDistributions|continuous-infinite}} {{DEFAULTSORT:Cauchy Distribution}} [[Category:Augustin-Louis Cauchy]] [[Category:Continuous distributions]] [[Category:Probability distributions with non-finite variance]] [[Category:Power laws]] [[Category:Stable distributions]] [[Category:Location-scale family probability distributions]] </textarea><div class="templatesUsed"><div class="mw-templatesUsedExplanation"><p><span id="templatesused">Pages transcluded onto the current version of this page<span class="posteditwindowhelplinks"> (<a 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