Cauchy distribution

Not to be confused with Lorenz curve.

Cauchy–Lorentz

Probability density function The purple curve is the standard Cauchy distribution
Cumulative distribution function
Parameters	$x_{0}\!$ location (real) $\displaystyle \gamma >0\!$ scale (real)
Support	$\displaystyle x\in (-\infty ,+\infty )\!$
PDF	${\frac {1}{\pi \gamma \,\left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}\!$
CDF	${\frac {1}{\pi }}\arctan \left({\frac {x-x_{0}}{\gamma }}\right)+{\frac {1}{2}}\!$
Mean	does not exist
Median	$x_{0}\!$
Mode	$x_{0}\!$
Variance	infinite
Skewness	does not exist
Excess kurtosis	does not exist
Entropy	$\log(\gamma )\,+\,\log(4\,\pi )\!$
MGF	does not exist
CF	$\displaystyle \exp(x_{0}\,i\,t-\gamma \,|t|)\!$

The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution.

Its importance in physics is the result of its being the solution to the differential equation describing forced resonance. In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane. In spectroscopy, it is the description of 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, and Chantler–Alda radiation. In its standard form, it is the maximum entropy probability distribution for a random variate X for which $E(\ln(1+X^{2}))=\ln(4)\,\!$.

Characterization

Probability density function

The Cauchy distribution has the probability density function

$f(x;x_{0},\gamma )={\frac {1}{\pi \gamma \left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}={1 \over \pi }\left[{\gamma \over (x-x_{0})^{2}+\gamma ^{2}}\right],$

where $x_{0}\,\!$ is the location parameter, specifying the location of the peak of the distribution, and $\gamma \,\!$ is the scale parameter which specifies the half-width at half-maximum (HWHM). $\gamma \,\!$ is also equal to half the interquartile range and is sometimes called the probable error. Augustin-Louis Cauchy exploited such a density function in 1827 with an infinitesimal scale parameter, defining what would now be called a Dirac delta function.

The amplitude of the above Lorentzian function is given by

${\text{Amplitude (or height)}}={\frac {1}{\pi \gamma }}.$

The special case when $x_{0}=0\,\!$ and $\gamma =1\,\!$ is called the standard Cauchy distribution with the probability density function

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

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

$f(x;x_{0},\gamma ,I)={\frac {I}{\left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}={I}\left[{\gamma ^{2} \over (x-x_{0})^{2}+\gamma ^{2}}\right],$

where $I\,\!$ is the height of the peak.

Cumulative distribution function

The cumulative distribution function (cdf) is:

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

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

$Q(p;x_{0},\gamma )=x_{0}+\gamma \,\tan \left[\pi \left(p-{\tfrac {1}{2}}\right)\right].$

It follows that the first and third quartiles are $\left(x_{0}-\gamma ,x_{0}+\gamma \right)\,\!$, and hence the interquartile range is $2\gamma \,\!$.

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

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

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

$h_{e}({\text{Cauchy}}(\gamma ))=\int _{0}^{1}\log \,(Q'(p;\gamma ))\,\mathrm {d} p=\log(\gamma )\,+\,\log(4\,\pi ).\!$

Properties

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 x₀.

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.

If $X_{1},\cdots ,X_{n}\,\!$ are independent and identically distributed random variables, each with a standard Cauchy distribution, then the sample mean $\left(X_{1}+\cdots +X_{n}\right)/n\,\!$ has the same standard Cauchy distribution (the sample median, which is not affected by extreme values, can be used as a measure of central tendency). To see that this is true, compute the characteristic function of the sample mean:

$\phi _{\overline {X}}(t)=\mathrm {E} \left(e^{i\,{\overline {X}}\,t}\right)\,\!$

where ${\overline {X}}$ is the sample mean. This example serves to show that the hypothesis 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.

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

The standard Cauchy distribution coincides with the Student's t-distribution with one degree of freedom.

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 Cauchy distribution is the only univariate distribution which is closed under linear fractional transformations with real coefficients. In this connection, see also McCullagh's parametrization of the Cauchy distributions.

Characteristic function

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

$\phi _{X}(t;x_{0},\gamma )=\mathrm {E} (e^{i\,X\,t})=\int _{-\infty }^{\infty }f(x;x_{0},\gamma )e^{i\,x\,t}\,dx=e^{ix_{0}t-\gamma \left|t\right|}.$

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:

$f(x;x_{0},\gamma )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\phi _{X}(t;x_{0},\gamma )e^{-i\,x\,t}\,dt\!$

Observe that the characteristic function is not differentiable at the origin: this corresponds to the fact that the Cauchy distribution does not have an expected value.

Explanation of undefined moments

Mean

If a probability distribution has a density function f(x), then the mean is

$\int _{-\infty }^{\infty }xf(x)\,dx.\qquad \qquad (1)\!$

The question is now whether this is the same thing as

$\int _{0}^{\infty }xf(x)\,dx-\int _{-\infty }^{0}|{x}|f(x)\,dx.\qquad \qquad (2)\!$

If at most one of the two terms in (2) is infinite, then (1) is the same as (2). But in the case of the Cauchy distribution, both the positive and negative terms of (2) are infinite. This means (2) is undefined. Moreover, if (1) is construed as a Lebesgue integral, then (1) is also undefined, because (1) is then defined simply as the difference (2) between positive and negative parts.

However, if (1) is construed as an improper integral rather than a Lebesgue integral, then (2) is undefined, and (1) is not necessarily well-defined. We may take (1) to mean

$\lim _{a\to \infty }\int _{-a}^{a}xf(x)\,dx,\!$

and this is its Cauchy principal value, which is zero, but we could also take (1) to mean, for example,

$\lim _{a\to \infty }\int _{-2a}^{a}xf(x)\,dx,\!$

which is not zero, as can be seen easily by computing the integral.

Because the integrand is bounded and is not Lebesgue integrable, it is not even Henstock–Kurzweil integrable. Various results in probability theory about expected values, such as the strong law of large numbers, will not work in such cases.

Higher moments

The Cauchy distribution does not have moments of any order. This follows from Hölder's inequality which implies that higher moments diverge if lower moments do. In particular, no second central moment exists, as can be verified by direct computation:

$\mathrm {E} (X^{2})\propto \int _{-\infty }^{\infty }{x^{2} \over 1+x^{2}}\,dx=\int _{-\infty }^{\infty }dx-\int _{-\infty }^{\infty }{1 \over 1+x^{2}}\,dx=\int _{-\infty }^{\infty }dx-\pi =\infty .\!$

The variance does not exist because of the divergent mean, which is distinctly different from having an infinite variance.

Estimation of parameters

Because the mean and variance of the Cauchy distribution are not defined, attempts to estimate these parameters will not be successful. For example, if N samples are taken from a Cauchy distribution, one may calculate the sample mean as:

${\overline {x}}={\frac {1}{N}}\sum _{i=1}^{N}x_{i}$

Although the sample values $\,\!x_{i}$ will be concentrated about the central value $\,\!x_{0}$, the sample mean will become increasingly variable as more samples are taken, because of the increased likelihood 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 samples 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 samples are taken.

Therefore, more robust means of estimating the central value $\,\!x_{0}$ and the scaling parameter $\,\!\gamma$ 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 $\,\!\gamma$. Other, more precise and robust methods have been developed 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. 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 $\,\!\gamma$. 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. Also, while the maximum likelihood estimator is asymptotically efficient, it is relatively inefficient for small samples. The log-likelihood function for the Cauchy distribution for sample size n is:

${\hat {\ell }}(\!x_{0},\gamma |\,x_{1},\dots ,x_{n})=n\log(\gamma )-\sum _{i=1}^{n}(\log[(\gamma )^{2}+(x_{i}-\!x_{0})^{2}])-n\log(\pi )$

Maximizing the log likelihood function with respect to $\,\!x_{0}$ and $\,\!\gamma$ produces the following system of equations:

$\sum _{i=1}^{n}(x_{i}-\!x_{0})/(\gamma ^{2}+[x_{i}-\!x_{0}]^{2})=0$

$\sum _{i=1}^{n}\gamma ^{2}/(\gamma ^{2}+[x_{i}-\!x_{0}]^{2})-n/2=0$

Note that $\sum _{i=1}^{n}\gamma ^{2}/(\gamma ^{2}+[x_{i}-\!x_{0}]^{2})$ is a monotone function in $\,\!\gamma$ and that the solution $\,\!\gamma$ must satisfy $\min |x_{i}-x_{0}|\leq \gamma \leq \max |x_{i}-x_{0}|$. Solving just for $\,\!x_{0}$ requires solving a polynomial of degree 2n − 1, and solving just for $\,\!\gamma$ requires solving a polynomial of degree $n$ (first for $\,\!\gamma ^{2}$, then $\,\!x_{0}$). Therefore, whether solving for one parameter or for both paramters 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. 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 $\,\!x_{0}$.

Circular Cauchy distribution

If X is Cauchy distributed with median μ and scale parameter γ, then the complex variable

$Z=(X-i)/(X+i)\,$

has unit modulus and is distributed on the unit circle with density:

$P_{cc}(\theta ;\zeta )={\frac {1-|\zeta |^{2}}{2\pi |e^{i\theta }-\zeta |^{2}}}$

with respect to the angular variable $\theta =\mathrm {arg} (z)$, where

$\zeta ={\frac {\psi -i}{\psi +i}}$

and $\psi$ expresses the two parameters of the associated linear Cauchy distribution for x as a complex number:

$\psi =\mu +i\gamma \,$

The distribution $P_{cc}(\theta ;\zeta )$ is called the circular Cauchy distribution (also the complex Cauchy distribution) with parameter $\zeta$. The circular Cauchy distribution is related to the wrapped Cauchy distribution. If $P_{wc}(\theta ;\psi )$ is a wrapped Cauchy distribution with the parameter $\psi =\mu +i\gamma$ representing the parameters of the corresponding "unwrapped" Cauchy distribution in the variable y where $\theta =(y\,\,\mathrm {mod} \,\,2\pi )$, then

$P_{wc}(\theta ;\psi )=P_{cc}(\theta ,e^{i\psi })\,$

See also McCullagh's parametrization of the Cauchy distributions and Poisson kernel for related concepts.

The circular Cauchy distribution expressed in complex form has finite moments of all orders

$\operatorname {E} (Z^{r})=\zeta ^{r},\quad \operatorname {E} ({\bar {Z}}^{r})={\bar {\zeta }}^{r}$

for integer $r\geq 1$. For $|\phi |<1$, the transformation

$U(z,\phi )=(z-\phi )/(1-{\bar {\phi }}z)$

is holomorphic on the unit disk, and the transformed variable $U(Z,\phi )$ is distributed as complex Cauchy with parameter $U(\zeta ,\phi )$.

Given a sample $z_{1},\ldots ,z_{n}$ of size n > 2, the maximum-likelihood equation

$n^{-1}U(z,{\hat {\zeta }})=n^{-1}\sum U(z_{j},{\hat {\zeta }})=0$

can be solved by a simple fixed-point iteration:

$\zeta ^{(r+1)}=U(n^{-1}U(z,\zeta ^{(r)}),-\zeta ^{(r)})\,$

starting with $\zeta ^{(0)}=0.$ The sequence of likelihood values is non-decreasing, and the solution is unique for samples containing at least three distinct values.

The maximum-likelihood estimate for the median (${\hat {\mu }}$) and scale parameter (${\hat {\gamma }}$) of a real Cauchy sample is obtained by the inverse transformation:

${\hat {\mu }}\pm i{\hat {\gamma }}=i(1+{\hat {\zeta }})/(1-{\hat {\zeta }}).$

For n ≤ 4, closed-form expressions are known for ${\hat {\zeta }}$. The density of the maximum-likelihood estimator at $t$ in the unit disk is necessarily of the form:

${\frac {p_{n}(\chi (t,\zeta ))}{4\pi (1-|t|^{2})^{2}}},$

where

$\chi (t,\zeta )={\frac {|t-\zeta |^{2}}{4(1-|t|^{2})(1-|\zeta |^{2})}}$.

Formulae for $p_{3}$ and $p_{4}$ are available.

Multivariate Cauchy distribution

A random vector X = (X₁, ..., X_k)′ is said to have the multivariate Cauchy distribution if every linear combination of its components Y = a₁X₁ + ... + a_kX_k has a Cauchy distribution. That is, for any constant vector a ∈ R^k, the random variable Y = a′X should have a univariate Cauchy distribution. The characteristic function of a multivariate Cauchy distribution is given by:

$\phi _{X}(t)=e^{i\,x_{0}\,(t)-\gamma \,(t)},\!$

where $\,x_{0}\,(t)$ and $\gamma \,(t)$ are real functions with $\,x_{0}\,(t)$ a homogeneous function of degree one and $\gamma \,(t)$ a positive homogeneous function of degree one. More formally:

$\,x_{0}\,(at)=a\,x_{0}\,(t)$ and $\gamma \,(at)=|a|\gamma \,(t)$ for all t.

An example of a bivariate Cauchy distribution can be given by:

$f(x,y;x_{0},y_{0},\gamma )={1 \over \pi }\left[{\gamma \over ((x-x_{0})^{2}+(y-y_{0})^{2}+\gamma ^{2})^{1.5}}\right].$

Note that in this example, even though there is no analogue to a covariance matrix, x and y are not statistically independent.

Analogously to the univariate density, the multidimensional Cauchy density also relates to the Multivariate Student distribution. They are equivalent when the degrees of freedom parameter is equal to one. The density of a k dimension Student distribution with one degree of freedom becomes:

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

Properties and details for this density can be obtained by taking it as a particular case of the Multivariate Student density.

Transformation properties

• If $X\sim {\textrm {Cauchy}}(x_{0},\gamma )\,$ then $kX+loc\sim {\textrm {Cauchy}}(x_{0}{k}+loc,\gamma k)\,$
• If $X\sim {\textrm {Cauchy}}(0,1)\,$ then ${\tfrac {2X}{1-X^{2}}}\sim {\textrm {Cauchy}}(0,1)\,$
• If $X\sim {\textrm {Cauchy}}(x_{0},\gamma _{0})\,$ and $Y\sim {\textrm {Cauchy}}(x_{1},\gamma _{1})\,$ then $X+Y\sim {\textrm {Cauchy}}(x_{0}+x_{1},\gamma _{0}+\gamma _{1})\,$
• If $X\sim {\textrm {Cauchy}}(0,\gamma )\,$ then ${\tfrac {1}{X}}\sim {\textrm {Cauchy}}(0,{\tfrac {1}{\gamma }})\,$
• McCullagh's parametrization of the Cauchy distributions: Expressing a Cauchy distribution in terms of one complex parameter $\psi =x_{0}+i\gamma$, define X ~ Cauchy$(\psi )$ to mean X ~ Cauchy$(x_{0},|\gamma |)$. If X ~ Cauchy$(\psi )$ then:

${\frac {aX+b}{cX+d}}$ ~ Cauchy$\left({\frac {a\psi +b}{c\psi +d}}\right)$ where a,b,c and d are real numbers.

• Using the same convention as above, if If X ~ Cauchy$(\psi )$ then:

${\frac {X-i}{X+i}}$ ~ CCauchy$\left({\frac {\psi -i}{\psi +i}}\right)$

where "CCauchy" is the circular Cauchy distribution.

Related distributions

• ${\textrm {Cauchy}}(0,1)\sim {\textrm {t}}(df=1)\,$ Student's t distribution
• If $X\sim {\textrm {N}}(0,1)\,$ and $Y\sim {\textrm {N}}(0,1)\,$ then ${\tfrac {X}{Y}}\sim {\textrm {Cauchy}}(0,1)\,$
• If $X\sim {\textrm {U}}(0,1)\,$ then $\tan \left({\pi \left(x-{\tfrac {1}{2}}\right)}\right)\sim {\textrm {Cauchy}}(0,1)\,$
• If $X\sim {\textrm {Log-Cauchy}}(0,1)\,$ then $\ln x\sim {\textrm {Cauchy}}(0,1)\,$
• The Cauchy distribution is a limiting case of a Pearson distribution of type 4
• The Cauchy distribution is a special case of a Pearson distribution of type 7
• The Cauchy distribution is a stable distribution: if X ~ Stable$(1,0,\gamma ,\mu )$, then X ~Cauchy(μ, γ).
• The Cauchy distribution is a singular limit of a Hyperbolic distribution
• The wrapped Cauchy distribution, taking values on a circle, is derived from the Cauchy distribution by wrapping it around the circle.

Relativistic Breit–Wigner distribution

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.

See also

• Lévy flight and Lévy process
• Slash distribution

External links

• Earliest Uses: The entry on Cauchy distribution has some historical information.
• Weisstein, Eric W. "Cauchy Distribution". MathWorld.
• GNU Scientific Library – Reference Manual