Cauchy distribution#Entropy

Here are the most important constructions.

If one stands in front of a line and kicks a ball with at a uniformly distributed random angle towards the line, then the distribution of the point where the ball hits the line is a Cauchy distribution.

For example, consider a point at $(x\_0,\; \backslash gamma)$ in the x-y plane, and select a line passing through the point, with its direction (angle with the $x$-axis) chosen uniformly (between −180° and 0°) at random. The intersection of the line with the x-axis follows a Cauchy distribution with location $x\_0$ and scale $\backslash gamma$.

This definition gives a simple way to sample from the standard Cauchy distribution. Let $u$ be a sample from a uniform distribution from $[0,1]$, then we can generate a sample, $x$ from the standard Cauchy distribution using

$$x\; =\; \backslash tan\backslash left(\backslash pi(u-\backslash tfrac\{1\}\{2\})\backslash right)$$

When $U$ and $V$ are two independent normally distributed random variables with expected value 0 and variance 1, then the ratio $U/V$ has the standard Cauchy distribution.

More generally, if $(U,\; V)$ is a rotationally symmetric distribution on the plane, then the ratio $U/V$ has the standard Cauchy distribution.

The Cauchy distribution is the probability distribution with the following probability density function (PDF){{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}}

$$f(x;\; x\_0,\backslash gamma)\; =\; \backslash frac\{1\}\{\backslash pi\backslash gamma\; \backslash left[1\; +\; \backslash left(\backslash frac\{x\; -\; x\_0\}\{\backslash gamma\}\backslash right)^2\backslash right]\}\; =\; \{\; 1\; \backslash over\; \backslash pi\; \}\; \backslash left[\; \{\; \backslash gamma\; \backslash over\; (x\; -\; x\_0)^2\; +\; \backslash gamma^2\; \}\; \backslash right],$$

where $x\_0$ is the location parameter, specifying the location of the peak of the distribution, and $\backslash gamma$ is the scale parameter which specifies the half-width at half-maximum (HWHM), alternatively $2\backslash gamma$ is full width at half maximum (FWHM). $\backslash gamma$ is also equal to half the interquartile range and is sometimes called the probable error. This function is also known as a Lorentzian function,{{cite web |title=Lorentzian Function |url=https://mathworld.wolfram.com/LorentzianFunction.html |website=MathWorld |publisher=Wolfram Research |accessdate=27 October 2024}} and an example of a nascent delta function, and therefore approaches a Dirac delta function in the limit as $\backslash gamma\; \backslash to\; 0$. Augustin-Louis Cauchy exploited such a density function in 1827 with an infinitesimal scale parameter, defining this Dirac delta function.

The maximum value or amplitude of the Cauchy PDF is $\backslash frac\{1\}\{\backslash pi\; \backslash gamma\}$, located at $x=x\_0$.

It is sometimes convenient to express the PDF in terms of the complex parameter $\backslash psi=\; x\_0\; +\; i\backslash gamma$

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)

The special case when $x\_0\; =\; 0$ and $\backslash gamma\; =\; 1$ is called the standard Cauchy distribution with the probability density function{{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}}{{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}}

$$f(x;\; 0,1)\; =\; \backslash frac\{1\}\{\backslash pi\; \backslash left(1\; +\; x^2\backslash right)\}.$$

In physics, a three-parameter Lorentzian function is often used:

$$f(x;\; x\_0,\backslash gamma,I)\; =\; \backslash frac\{I\}\{\backslash left[1\; +\; \{\backslash left(\backslash frac\{x-x\_0\}\{\backslash gamma\}\backslash right)\}^2\backslash right]\}\; =\; I\; \backslash left[\; \backslash frac\{\backslash gamma^2\}\{\{\backslash left(x\; -\; x\_0\backslash right)\}^2\; +\; \backslash gamma^2\; \}\; \backslash right],$$

where $I$ 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 $I\; =\; \backslash frac\{1\}\{\backslash pi\backslash gamma\}.\backslash !$

The Cauchy distribution is the probability distribution with the following cumulative distribution function (CDF):

$$F(x;\; x\_0,\backslash gamma)=\backslash frac\{1\}\{\backslash pi\}\; \backslash arctan\backslash left(\backslash frac\{x-x\_0\}\{\backslash gamma\}\backslash right)+\backslash frac\{1\}\{2\}$$

and the quantile function (inverse cdf) of the Cauchy distribution is

$$Q(p;\; x\_0,\backslash gamma)\; =\; x\_0\; +\; \backslash gamma\backslash ,\backslash tan\backslash left[\backslash pi\backslash left(p-\backslash tfrac\{1\}\{2\}\backslash right)\backslash right].$$

It follows that the first and third quartiles are $(x\_0\; -\; \backslash gamma,\; x\_0\; +\; \backslash gamma)$, and hence the interquartile range is $2\backslash gamma$.

For the standard distribution, the cumulative distribution function simplifies to arctangent function $\backslash arctan(x)$:

$$F(x;\; 0,1)=\backslash frac\{1\}\{\backslash pi\}\; \backslash arctan\backslash left(x\backslash right)+\backslash frac\{1\}\{2\}$$

The standard Cauchy distribution is the 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.{{Cite journal |last1=Li |first1=Rui |last2=Nadarajah |first2=Saralees |date=2020-03-01 |title=A review of Student's t distribution and its generalizations |url=https://pure.manchester.ac.uk/ws/files/78743532/studentt.pdf |journal=Empirical Economics |language=en |volume=58 |issue=3 |pages=1461–1490 |doi=10.1007/s00181-018-1570-0 |issn=1435-8921}}

If $\backslash Sigma$ is a $p\backslash times\; p$ positive-semidefinite covariance matrix with strictly positive diagonal entries, then for independent and identically distributed $X,Y\backslash sim\; N(0,\backslash Sigma)$ and any random $p$-vector $w$ independent of $X$ and $Y$ such that $w\_1+\backslash cdots+w\_p=1$ and $w\_i\backslash geq\; 0,\; i=1,\backslash ldots,p,$ (defining a categorical distribution) it holds that{{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}}

$$\backslash sum\_\{j=1\}^p\; w\_j\backslash frac\{X\_j\}\{Y\_j\}\backslash sim\backslash mathrm\{Cauchy\}(0,1).$$

The Cauchy distribution is an example of a distribution which has no mean, variance or higher moments defined. Its mode and median are well defined and are both equal to $x\_0$.

The Cauchy distribution is an infinitely divisible probability distribution. It is also a strictly stable distribution.{{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 }}

Like all stable distributions, the location-scale family to which the Cauchy distribution belongs is closed under linear transformations with real coefficients. In addition, the family of Cauchy-distributed random variables is closed under linear fractional transformations with real coefficients.{{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}} In this connection, see also McCullagh's parametrization of the Cauchy distributions.

If $X\_1,\; X\_2,\; \backslash ldots,\; X\_n$ are an IID sample from the standard Cauchy distribution, then their sample mean $\backslash bar\; X\; =\; \backslash frac\; 1\; n\; \backslash sum\_i\; X\_i$ 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 of the standard Cauchy distribution (see below):$$\backslash varphi\_X(t)\; =\; \backslash operatorname\{E\}\backslash left[e^\{iXt\}\; \backslash right\; ]\; =\; e^\{-|t|\}.$$With this, we have $\backslash varphi\_\{\backslash sum\_i\; X\_i\}(t)\; =\; e^\{-n\; |t|\}$, and so $\backslash bar\; X$ has a standard Cauchy distribution.

More generally, if $X\_1,\; X\_2,\; \backslash ldots,\; X\_n$ are independent and Cauchy distributed with location parameters $x\_1,\; \backslash ldots,\; x\_n$ and scales $\backslash gamma\_1,\; \backslash ldots,\; \backslash gamma\_n$, and $a\_1,\; \backslash ldots,\; a\_n$ are real numbers, then $\backslash sum\_i\; a\_i\; X\_i$ is Cauchy distributed with location $\backslash sum\_i\; a\_i\; x\_i$ and scale$\backslash sum\_i\; |a\_i|\; \backslash gamma\_i$. 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 distributions, of which the Cauchy distribution is a special case.

If $X\_1,\; X\_2,\; \backslash ldots$ are an IID sample with PDF $\backslash rho$ such that $\backslash lim\_\{c\; \backslash to\; \backslash infty\}\; \backslash frac\{1\}\{c\}\; \backslash int\_\{-c\}^c\; x^2\; \backslash rho(x)\; \backslash ,\; dx\; =\; \backslash frac\{2\backslash gamma\}\{\backslash pi\}$ is finite, but nonzero, then $\backslash frac\; 1n\; \backslash sum\_\{i=1\}^n\; X\_i$ converges in distribution to a Cauchy distribution with scale $\backslash gamma$.{{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}}

Let $X$ denote a Cauchy distributed random variable. The characteristic function of the Cauchy distribution is given by

$$\backslash varphi\_X(t)\; =\; \backslash operatorname\{E\}\backslash left[e^\{iXt\}\; \backslash right\; ]\; =\backslash int\_\{-\backslash infty\}^\backslash infty\; f(x;x\_0,\backslash gamma)e^\{ixt\}\backslash ,dx\; =\; e^\{ix\_0t\; -\; \backslash gamma\; |t|\}.$$

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:

$$f(x;\; x\_0,\backslash gamma)\; =\; \backslash frac\{1\}\{2\backslash pi\}\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash varphi\_X(t;x\_0,\backslash gamma)e^\{-ixt\}\; \backslash ,\; dt\; \backslash !$$

The nth moment of a distribution is the nth derivative of the characteristic function evaluated at $t=0$. Observe that the characteristic function is not differentiable at the origin: this corresponds to the fact that the Cauchy distribution does not have well-defined moments higher than the zeroth moment.

The Kullback–Leibler divergence between two Cauchy distributions has the following symmetric closed-form formula:{{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}}

\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}.

Any f-divergence between two Cauchy distributions is symmetric and can be expressed as a function of the chi-squared divergence.{{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 }}

Closed-form expression for the total variation, Jensen–Shannon divergence, Hellinger distance, etc. are available.

The entropy of the Cauchy distribution is given by:

\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}

The derivative of the quantile function, the quantile density function, for the Cauchy distribution is:

$$Q\text{'}(p;\; \backslash gamma)\; =\; \backslash gamma\; \backslash pi\; \backslash ,\; \backslash sec^2\backslash left[\backslash pi\backslash left(p\; -\; \backslash tfrac\{1\}\{2\}\backslash right)\backslash right].$$

The differential entropy of a distribution can be defined in terms of its quantile density,{{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 }} specifically:

$$H(\backslash gamma)\; =\; \backslash int\_0^1\; \backslash log\backslash ,(Q\text{'}(p;\; \backslash gamma))\backslash ,\backslash mathrm\; dp\; =\; \backslash log(4\backslash pi\backslash gamma)$$

The Cauchy distribution is the maximum entropy probability distribution for a random variate $X$ for which{{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}}

$$\backslash operatorname\{E\}\backslash left[\backslash log\backslash left(1\; +\; \{\backslash left(\backslash frac\{X-x\_0\}\{\backslash gamma\}\backslash right)\}^2\backslash right)\backslash right]\; =\; \backslash log\; 4$$

The Cauchy distribution is usually used as an illustrative counterexample in elementary probability courses, as a distribution with no well-defined (or "indefinite") moments.

If we take an IID sample $X\_1,\; X\_2,\; \backslash ldots$ from the standard Cauchy distribution, then the sequence of their sample mean is $S\_n\; =\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^n\; X\_i$, 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 $V\_n\; =\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^n\; \{\backslash left(X\_i\; -\; S\_n\backslash right)\}^2$ also does not converge.

File:Sample mean and variance of IID samples from a standard Cauchy distribution..png

A typical trajectory of $S\_1,\; S\_2,\; ...$ 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 $V\_1,\; V\_2,\; ...$ 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.

If a probability distribution has a density function $f(x)$, then the mean, if it exists, is given by

{{NumBlk||$$\backslash int\_\{-\backslash infty\}^\backslash infty\; x\; f(x)\backslash ,dx.$$|{{EquationRef|1}}}}

We may evaluate this two-sided improper integral by computing the sum of two one-sided improper integrals. That is,

{{NumBlk||$$\backslash int\_\{-\backslash infty\}^a\; x\; f(x)\backslash ,dx\; +\backslash int\_a^\backslash infty\; x\; f(x)\; \backslash ,\; dx$$|{{EquationRef|2}}}}

for an arbitrary real number $a$.

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.{{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 }} 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

$$\backslash lim\_\{a\backslash to\backslash infty\}\backslash int\_\{-a\}^a\; x\; f(x)\backslash ,dx$$

which is zero. On the other hand, the related integral

$$\backslash lim\_\{a\backslash to\backslash infty\}\backslash int\_\{-2a\}^a\; x\; f(x)\backslash ,dx$$

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 values, such as the strong law of large numbers, fail to hold for the Cauchy distribution.

The absolute moments for $p\backslash in(-1,1)$ are defined.

For $X\backslash sim\backslash mathrm\{Cauchy\}(0,\backslash gamma)$ we have

$$\backslash operatorname\{E\}[|X|^p]\; =\; \backslash gamma^p\; \backslash mathrm\{sec\}(\backslash pi\; p/2).$$

The Cauchy distribution does not have finite moments of any order. Some of the higher raw moments do exist and have a value of infinity, for example, the raw second moment:

$$\backslash begin\{align\}$$

\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}

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 $\backslash infty\; -\; \backslash infty$ 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 moments and standardized moments 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.

Consider the truncated distribution defined by restricting the standard Cauchy distribution to the interval {{math|[−10¹⁰⁰, 10¹⁰⁰]}}. 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.{{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 }}.

• If $X\; \backslash sim\; \backslash operatorname\{Cauchy\}(x\_0,\backslash gamma)$ then $kX\; +\; \backslash ell\; \backslash sim\; \backslash textrm\{Cauchy\}(x\_0\; k+\backslash ell,\; \backslash gamma\; |k|)${{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

}}

• If $X\; \backslash sim\; \backslash operatorname\{Cauchy\}(x\_0,\; \backslash gamma\_0)$ and $Y\; \backslash sim\; \backslash operatorname\{Cauchy\}(x\_1,\backslash gamma\_1)$ are independent, then $X+Y\; \backslash sim\; \backslash operatorname\{Cauchy\}(x\_0+x\_1,\backslash gamma\_0\; +\backslash gamma\_1)$ and $X-Y\; \backslash sim\; \backslash operatorname\{Cauchy\}(x\_0-x\_1,\; \backslash gamma\_0+\backslash gamma\_1)$
• If $X\; \backslash sim\; \backslash operatorname\{Cauchy\}(0,\backslash gamma)$ then $\backslash tfrac\{1\}\{X\}\; \backslash sim\; \backslash operatorname\{Cauchy\}(0,\; \backslash tfrac\{1\}\{\backslash gamma\})$
• McCullagh's parametrization of the Cauchy distributions: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 247–259. [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. Expressing a Cauchy distribution in terms of one complex parameter $\backslash psi\; =\; x\_0+i\backslash gamma$, define $X\; \backslash sim\; \backslash operatorname\{Cauchy\}(\backslash psi)$ to mean $X\; \backslash sim\; \backslash operatorname\{Cauchy\}(x\_0,|\backslash gamma|)$. If $X\; \backslash sim\; \backslash operatorname\{Cauchy\}(\backslash psi)$ then: $$\backslash frac\{aX+b\}\{cX+d\}\; \backslash sim\; \backslash operatorname\{Cauchy\}\backslash left(\backslash frac\{a\backslash psi+b\}\{c\backslash psi+d\}\backslash right)$$ where $a$, $b$, $c$ and $d$ are real numbers.
• Using the same convention as above, if $X\; \backslash sim\; \backslash operatorname\{Cauchy\}(\backslash psi)$ then: $$\backslash frac\{X-i\}\{X+i\}\; \backslash sim\; \backslash operatorname\{CCauchy\}\backslash left(\backslash frac\{\backslash psi-i\}\{\backslash psi+i\}\backslash right)$$where $\backslash operatorname\{CCauchy\}$ is the circular Cauchy distribution.

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.{{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}} For example, if an i.i.d. sample of size n is taken from a Cauchy distribution, one may calculate the sample mean as:

$$\backslash bar\{x\}=\backslash frac\; 1\; n\; \backslash sum\_\{i=1\}^n\; x\_i$$

Although the sample values $x\_i$ will be concentrated about the central value $x\_0$, 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 $x\_0$ 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 $x\_0$ and the scaling parameter $\backslash gamma$ are needed. One simple method is to take the median value of the sample as an estimator of $x\_0$ and half the sample interquartile range as an estimator of $\backslash gamma$. Other, more precise and robust methods have been developed {{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}}{{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 }} For example, the truncated mean of the middle 24% of the sample order statistics produces an estimate for $x\_0$ that is more efficient than using either the sample median or the full sample mean.{{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}}{{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=852–855|jstor=2282794|doi=10.1080/01621459.1966.10480912}} 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.

Maximum likelihood can also be used to estimate the parameters $x\_0$ and $\backslash gamma$. 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.{{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}} Also, while the maximum likelihood estimator is asymptotically efficient, it is relatively inefficient for small samples.{{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}}{{cite book|title=Introduction to Robust Estimation & Hypothesis Testing |last1=Wilcox |first1=Rand |year=2012 |publisher=Elsevier}} The log-likelihood function for the Cauchy distribution for sample size $n$ is:

$$\backslash hat\backslash ell(x\_1,\backslash dotsc,x\_n\; \backslash mid\; \backslash !x\_0,\backslash gamma\; )\; =\; -\; n\; \backslash log\; (\backslash gamma\; \backslash pi)\; -\; \backslash sum\_\{i=1\}^n\; \backslash log\; \backslash left(1\; +\; \backslash left(\backslash frac\{x\_i\; -\; x\_0\}\{\backslash gamma\}\backslash right)^2\backslash right)$$

Maximizing the log likelihood function with respect to $x\_0$ and $\backslash gamma$ by taking the first derivative produces the following system of equations:

$$\backslash frac\{d\; \backslash ell\}\{d\; x\_\{0\}\}\; =\; \backslash sum\_\{i=1\}^n\; \backslash frac\{2(x\_i\; -\; x\_0)\}\{\backslash gamma^2\; +\; \backslash left(x\_i\; -\; \backslash !x\_0\backslash right)^2\}\; =0$$

$$\backslash frac\{d\; \backslash ell\}\{d\; \backslash gamma\}\; =\; \backslash sum\_\{i=1\}^n\; \backslash frac\{2\backslash left(x\_i\; -\; x\_0\backslash right)^2\}\{\backslash gamma\; (\backslash gamma^2\; +\; \backslash left(x\_i\; -\; x\_0\backslash right)^2)\}\; -\; \backslash frac\{n\}\{\backslash gamma\}\; =\; 0$$

Note that

$$\backslash sum\_\{i=1\}^n\; \backslash frac\{\backslash left(x\_i\; -\; x\_0\backslash right)^2\}\{\backslash gamma^2\; +\; \backslash left(x\_i\; -\; x\_0\backslash right)^2\}$$

is a monotone function in $\backslash gamma$ and that the solution $\backslash gamma$ must satisfy

$$\backslash min\; |x\_i-x\_0|\backslash le\; \backslash gamma\backslash le\; \backslash max\; |x\_i-x\_0|.$$

Solving just for $x\_0$ requires solving a polynomial of degree $2n-1$, and solving just for $\backslash ,\backslash !\backslash gamma$ requires solving a polynomial of degree $2n$. Therefore, whether solving for one parameter or for both parameters simultaneously, a numerical solution on a computer is typically required. The benefit of maximum likelihood estimation is asymptotic efficiency; estimating $x\_0$ using the sample median is only about 81% as asymptotically efficient as estimating $x\_0$ by maximum likelihood.{{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}} The truncated sample mean using the middle 24% order statistics is about 88% as asymptotically efficient an estimator of $x\_0$ as the maximum likelihood estimate. 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 $x\_0$.

The shape can be estimated using the median of absolute values, since for location 0 Cauchy variables $X\backslash sim\backslash mathrm\{Cauchy\}(0,\backslash gamma)$, the $\backslash operatorname\{median\}(|X|)\; =\; \backslash gamma$ the shape parameter.

• $\backslash operatorname\{Cauchy\}(0,1)\; \backslash sim\; \backslash textrm\{t\}(\backslash mathrm\{df\}=1)\backslash ,$ Student's t distribution
• $\backslash operatorname\{Cauchy\}(\backslash mu,\backslash sigma)\; \backslash sim\; \backslash textrm\{t\}\_\{(\backslash mathrm\{df\}=1)\}(\backslash mu,\backslash sigma)\backslash ,$ non-standardized Student's t distribution
• If $X,\; Y\; \backslash sim\; \backslash textrm\{N\}(0,1)\backslash ,\; X,\; Y$ independent, then $\backslash tfrac\; X\; Y\backslash sim\; \backslash textrm\{Cauchy\}(0,1)\backslash ,$
• If $X\; \backslash sim\; \backslash textrm\{U\}(0,1)\backslash ,$ then $\backslash tan\; \backslash left(\; \backslash pi\; \backslash left(X-\backslash tfrac\{1\}\{2\}\backslash right)\; \backslash right)\; \backslash sim\; \backslash textrm\{Cauchy\}(0,1)\backslash ,$
• If $X\; \backslash sim\; \backslash operatorname\{Log-Cauchy\}(0,\; 1)$ then $\backslash ln(X)\; \backslash sim\; \backslash textrm\{Cauchy\}(0,\; 1)$
• If $X\; \backslash sim\; \backslash operatorname\{Cauchy\}(x\_0,\backslash gamma)$ then $\backslash tfrac1X\; \backslash sim\; \backslash operatorname\{Cauchy\}\backslash left(\backslash tfrac\{x\_0\}\{x\_0^2+\backslash gamma^2\},\backslash tfrac\{\backslash gamma\}\{x\_0^2+\backslash gamma^2\}\backslash right)$
• 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.
• The Cauchy distribution is a stable distribution: if $X\; \backslash sim\; \backslash textrm\{Stable\}(1,\; 0,\; \backslash gamma,\; \backslash mu)$, then $X\; \backslash sim\; \backslash operatorname\{Cauchy\}(\backslash mu,\; \backslash gamma)$.
• 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 $X\; \backslash sim\; \backslash textrm\{N\}(0,1)$, $Z\; \backslash sim\; \backslash operatorname\{Inverse-Gamma\}(1/2,\; s^2/2)$, then $Y\; =\; \backslash mu\; +\; X\; \backslash sqrt\; Z\; \backslash sim\; \backslash operatorname\{Cauchy\}(\backslash mu,s)$. For half-Cauchy distributions, the relation holds by setting $X\; \backslash sim\; \backslash textrm\{N\}(0,1)\; I\backslash \{X\backslash ge0\backslash \}$.

The Cauchy distribution is the stable distribution of index 1. The Lévy–Khintchine representation of such a stable distribution of parameter $\backslash gamma$ is given, for $X\; \backslash sim\; \backslash operatorname\{Stable\}(\backslash gamma,\; 0,\; 0)\backslash ,$ by:

$$\backslash operatorname\{E\}\backslash left(\; e^\{ixX\}\; \backslash right)\; =\; \backslash exp\backslash left(\; \backslash int\_\{\; \backslash mathbb\{R\}\; \}\; (e^\{ixy\}\; -\; 1)\; \backslash Pi\_\backslash gamma(dy)\; \backslash right)$$

where

and $c\_\{1,\; \backslash gamma\},\; c\_\{2,\; \backslash gamma\}$ can be expressed explicitly.{{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 }} In the case $\backslash gamma\; =\; 1$ of the Cauchy distribution, one has $c\_\{1,\; \backslash gamma\}\; =\; c\_\{2,\; \backslash gamma\}$.

This last representation is a consequence of the formula

$$\backslash pi\; |x|\; =\; \backslash operatorname\{PV\; \}\backslash int\_\{\backslash mathbb\{R\}\; \backslash smallsetminus\backslash lbrace\; 0\; \backslash rbrace\}\; (1\; -\; e^\{ixy\})\; \backslash ,\; \backslash frac\{dy\}\{y^2\}$$

A random vector $X=(X\_1,\; \backslash ldots,\; X\_k)^T$ is said to have the multivariate Cauchy distribution if every linear combination of its components $Y=a\_1X\_1+\; \backslash cdots\; +\; a\_kX\_k$ has a Cauchy distribution. That is, for any constant vector $a\backslash in\; \backslash mathbb\; R^k$, the random variable $Y=a^TX$ should have a univariate Cauchy distribution.{{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}} The characteristic function of a multivariate Cauchy distribution is given by:

$$\backslash varphi\_X(t)\; =\; e^\{ix\_0(t)-\backslash gamma(t)\},\; \backslash !$$

where $x\_0(t)$ and $\backslash gamma(t)$ are real functions with $x\_0(t)$ a homogeneous function of degree one and $\backslash gamma(t)$ a positive homogeneous function of degree one. More formally:

$$\backslash begin\{align\}$$

x_0(at) &= a x_0(t), \\

\gamma (at) &= |a| \gamma (t),

\end{align}

for all $t$.

An example of a bivariate Cauchy distribution can be given by:{{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=713–738|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}}

$$f(x,\; y;\; x\_0,y\_0,\backslash gamma)\; =\; \backslash frac\{1\}\{2\; \backslash pi\}\; \backslash ,\; \backslash frac\{\backslash gamma\}\{\{\backslash left(\{\backslash left(x\; -\; x\_0\backslash right)\}^2\; +\; \{\backslash left(y\; -\; y\_0\backslash right)\}^2\; +\; \backslash gamma^2\backslash right)\}^\{3/2\}\}\; .$$

Note that in this example, even though the covariance between $x$ and $y$ is 0, $x$ and $y$ are not statistically independent.

We also can write this formula for complex variable. Then the probability density function of complex Cauchy is :

$$f(z;\; z\_0,\backslash gamma)\; =\; \backslash frac\{1\}\{2\backslash pi\}\; \backslash ,\backslash frac\{\backslash gamma\}\{\{\backslash left(\{\backslash left|z\; -\; z\_0\backslash right|\}^2\; +\; \backslash gamma^2\backslash right)\}^\{3/2\}\; \}\; .$$

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 $k$ dimension Student distribution with one degree of freedom is:

$$f(\backslash mathbf\{x\};\; \backslash boldsymbol\{\backslash mu\},\backslash mathbf\{\backslash Sigma\},\; k)=\; \backslash frac\{\backslash Gamma\{\backslash left(\backslash frac\{1+k\}\{2\}\backslash right)\}\}\{\backslash Gamma(\backslash frac\{1\}\{2\})\; \backslash pi^\{\backslash frac\{k\}\{2\}\}\; \backslash left|\backslash mathbf\{\backslash Sigma\}\backslash right|^\{\backslash frac\{1\}\{2\}\}\; \backslash left[1\; +\; (\{\backslash mathbf\; x\}-\{\backslash boldsymbol\backslash mu\})^\backslash mathsf\{T\}\; \{\backslash mathbf\backslash Sigma\}^\{-1\}\; (\{\backslash mathbf\; x\}-\{\backslash boldsymbol\backslash mu\})\backslash right]^\{\backslash frac\{1+k\}\{2\}\}\}\; .$$

The properties of multidimensional Cauchy distribution are then special cases of the multivariate Student distribution.

File:Cauchy distribution.png, see also distribution fitting{{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}}]]

• In spectroscopy, the Cauchy distribution describes the shape of spectral lines 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 collision broadening.{{cite book |author=E. Hecht |year=1987 |title=Optics |page=603 |edition=2nd |publisher=Addison-Wesley }} 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 {{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}} derived the test statistic for estimators of $\backslash hat\{\backslash beta\}$ for the equation $x\_\{t+1\}=\backslash beta\{x\}\_t+\backslash varepsilon\_\{t+1\},\backslash beta>1$ and where the maximum likelihood estimator is found using ordinary least squares showed the sampling distribution of the statistic is the Cauchy distribution.
• 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 problemGull, 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 }} 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 positions as part of the cumulative frequency analysis.
• The expression for the imaginary part of complex 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.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 }}

{{Main article|Relativistic Breit–Wigner distribution}}

In 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}}

File:Mean estimator consistency.gif (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 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 Poisson in 1824, with Cauchy only becoming associated with it during an academic controversy in 1853.Cauchy and the Witch of Agnesi in Statistics on the Table, S M Stigler Harvard 1999 Chapter 18 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, 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 Bienaymé, who was to engage Cauchy in a long dispute over the matter.

• Lévy flight and Lévy process
• Laplace distribution, the Fourier transform of the Cauchy distribution
• Cauchy process
• Stable process
• Slash distribution

{{Reflist|30em}}

• {{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 – 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

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