Cauchy

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

The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution ${\displaystyle f(x;x_{0},\gamma )}$ is the distribution of the x-intercept of a ray issuing from ${\displaystyle (x_{0},\gamma )}$ with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero.

The Cauchy distribution is often used in statistics as the canonical example of a "pathological" distribution since both its expected value and its variance are undefined (but see § Moments below). The Cauchy distribution does not have finite moments of order greater than or equal to one; only fractional absolute moments exist.^[1] 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 distributions with a probability density function that can be expressed analytically, the others being the normal distribution and the Lévy distribution.

History

Estimating the mean and standard deviation through samples from a Cauchy distribution (bottom) does not converge with more samples, 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 Fermat in 1659, and later was known as the witch of Agnesi, after 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.^[2] Poisson noted that if the mean of observations following such a distribution were taken, the mean error 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.

Constructions

Here are the most important constructions.

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 ${\displaystyle x_{0}}$.

The Cauchy distribution is an infinitely divisible probability distribution. It is also a strictly stable distribution.^[7]

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 closed under linear fractional transformations with real coefficients.^[8] In this connection, see also McCullagh's parametrization of the Cauchy distributions.

Sum of Cauchy distributions

If ${\displaystyle X_{1},X_{2},...,X_{n}}$ are IID samples from the standard Cauchy distribution, then their sample mean ${\displaystyle {\bar {X}}={\frac {1}{n}}\sum _{i}X_{i}}$ 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 standard Cauchy distribution (see below):

$${\displaystyle \varphi _{X}(t)=\operatorname {E} \left[e^{iXt}\right]=e^{-|t|}.}$$

With this, we have ${\displaystyle \varphi _{\sum _{i}X_{i}}(t)=e^{-n|t|}}$, and so ${\displaystyle {\bar {X}}}$ has a standard Cauchy distribution.

More generally, if ${\displaystyle X_{1},X_{2},...,X_{n}}$ are independent and Cauchy distributed with location parameters ${\displaystyle x_{1},...,x_{n}}$ and scales ${\displaystyle \gamma _{1},...,\gamma _{n}}$, and ${\displaystyle a_{1},...,a_{n}}$ are real numbers, then ${\displaystyle \sum _{i}a_{i}X_{i}}$ is Cauchy distributed with location ${\displaystyle \sum _{i}a_{i}x_{i}}$ and scale${\displaystyle \sum _{i}|a_{i}|\gamma _{i}}$. 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.

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 IID samples ${\displaystyle X_{1},X_{2},..}$ from the standard Cauchy distribution, then the sequence of their sample mean is ${\displaystyle S_{n}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}$, 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 ${\displaystyle V_{n}={\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-S_{n})^{2}}$ also does not converge.

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 ${\displaystyle S_{1},S_{2},...}$ 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 ${\displaystyle V_{1},V_{2},...}$ 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.

Mean

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

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

(1)

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

$${\displaystyle \int _{-\infty }^{a}xf(x)\,dx+\int _{a}^{\infty }xf(x)\,dx}$$

(2)

for an arbitrary real number ${\displaystyle a}$.

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 (2) are infinite and have opposite sign. Hence (1) is undefined, and thus so is the mean.^[14]

Note that the Cauchy principal value of the mean of the Cauchy distribution is

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

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

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

is not zero, as can be seen by computing the integral. This again shows that the mean (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.^[14]

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.^[16] For example, if an i.i.d. sample of size n is taken from a Cauchy distribution, one may calculate the sample mean as:

${\displaystyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}$

Although the sample values ${\displaystyle x_{i}}$ will be concentrated about the central value ${\displaystyle x_{0}}$, 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 ${\displaystyle x_{0}}$ 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 ${\displaystyle x_{0}}$ and the scaling parameter ${\displaystyle \gamma }$ are needed. One simple method is to take the median value of the sample as an estimator of ${\displaystyle x_{0}}$ and half the sample interquartile range as an estimator of ${\displaystyle \gamma }$. Other, more precise and robust methods have been developed ^[17][18] For example, the truncated mean of the middle 24% of the sample order statistics produces an estimate for ${\displaystyle x_{0}}$ that is more efficient than using either the sample median or the full sample mean.^[19][20] 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.^[19][20]

Maximum likelihood can also be used to estimate the parameters ${\displaystyle x_{0}}$ and ${\displaystyle \gamma }$. 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.^[21] Also, while the maximum likelihood estimator is asymptotically efficient, it is relatively inefficient for small samples.^[22][23] The log-likelihood function for the Cauchy distribution for sample size ${\displaystyle n}$ is:

${\displaystyle {\hat {\ell }}(x_{1},\dotsc ,x_{n}\mid \!x_{0},\gamma )=-n\log(\gamma \pi )-\sum _{i=1}^{n}\log \left(1+\left({\frac {x_{i}-x_{0}}{\gamma }}\right)^{2}\right)}$

Maximizing the log likelihood function with respect to ${\displaystyle x_{0}}$ and ${\displaystyle \gamma }$ by taking the first derivative produces the following system of equations:

${\displaystyle {\frac {d\ell }{dx_{0}}}=\sum _{i=1}^{n}{\frac {2(x_{i}-x_{0})}{\gamma ^{2}+\left(x_{i}-\!x_{0}\right)^{2}}}=0}$

${\displaystyle {\frac {d\ell }{d\gamma }}=\sum _{i=1}^{n}{\frac {2\left(x_{i}-x_{0}\right)^{2}}{\gamma (\gamma ^{2}+\left(x_{i}-x_{0}\right)^{2})}}-{\frac {n}{\gamma }}=0}$

Note that

${\displaystyle \sum _{i=1}^{n}{\frac {\left(x_{i}-x_{0}\right)^{2}}{\gamma ^{2}+\left(x_{i}-x_{0}\right)^{2}}}}$

is a monotone function in ${\displaystyle \gamma }$ and that the solution ${\displaystyle \gamma }$ must satisfy

${\displaystyle \min |x_{i}-x_{0}|\leq \gamma \leq \max |x_{i}-x_{0}|.}$

Solving just for ${\displaystyle x_{0}}$ requires solving a polynomial of degree ${\displaystyle 2n-1}$,^[21] and solving just for ${\displaystyle \,\!\gamma }$ requires solving a polynomial of degree ${\displaystyle 2n}$. 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 ${\displaystyle x_{0}}$ using the sample median is only about 81% as asymptotically efficient as estimating ${\displaystyle x_{0}}$ by maximum likelihood.^[20][24] The truncated sample mean using the middle 24% order statistics is about 88% as asymptotically efficient an estimator of ${\displaystyle x_{0}}$ as the maximum likelihood estimate.^[20] 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 ${\displaystyle x_{0}}$.

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

Multivariate Cauchy distribution

A random vector ${\displaystyle X=(X_{1},\ldots ,X_{k})^{T}}$ is said to have the multivariate Cauchy distribution if every linear combination of its components ${\displaystyle Y=a_{1}X_{1}+\cdots +a_{k}X_{k}}$ has a Cauchy distribution. That is, for any constant vector ${\displaystyle a\in \mathbb {R} ^{k}}$, the random variable ${\displaystyle Y=a^{T}X}$ should have a univariate Cauchy distribution.^[25] The characteristic function of a multivariate Cauchy distribution is given by:

${\displaystyle \varphi _{X}(t)=e^{ix_{0}(t)-\gamma (t)},\!}$

where ${\displaystyle x_{0}(t)}$ and ${\displaystyle \gamma (t)}$ are real functions with ${\displaystyle x_{0}(t)}$ a homogeneous function of degree one and ${\displaystyle \gamma (t)}$ a positive homogeneous function of degree one.^[25] More formally:^[25]

${\displaystyle x_{0}(at)=ax_{0}(t),}$

${\displaystyle \gamma (at)=|a|\gamma (t),}$

for all ${\displaystyle t}$.

An example of a bivariate Cauchy distribution can be given by:^[26]

${\displaystyle f(x,y;x_{0},y_{0},\gamma )={1 \over 2\pi }\left[{\gamma \over ((x-x_{0})^{2}+(y-y_{0})^{2}+\gamma ^{2})^{3/2}}\right].}$

Note that in this example, even though the covariance between ${\displaystyle x}$ and ${\displaystyle y}$ is 0, ${\displaystyle x}$ and ${\displaystyle y}$ are not statistically independent.^[26]

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

${\displaystyle f(z;z_{0},\gamma )={1 \over 2\pi }\left[{\gamma \over (|z-z_{0}|^{2}+\gamma ^{2})^{3/2}}\right].}$

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 ${\displaystyle k}$ dimension Student distribution with one degree of freedom is:

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

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

Transformation properties

• If ${\displaystyle X\sim \operatorname {Cauchy} (x_{0},\gamma )}$ then ${\displaystyle kX+\ell \sim {\textrm {Cauchy}}(x_{0}k+\ell ,\gamma |k|)}$^[27]
• If ${\displaystyle X\sim \operatorname {Cauchy} (x_{0},\gamma _{0})}$ and ${\displaystyle Y\sim \operatorname {Cauchy} (x_{1},\gamma _{1})}$ are independent, then ${\displaystyle X+Y\sim \operatorname {Cauchy} (x_{0}+x_{1},\gamma _{0}+\gamma _{1})}$ and ${\displaystyle X-Y\sim \operatorname {Cauchy} (x_{0}-x_{1},\gamma _{0}+\gamma _{1})}$
• If ${\displaystyle X\sim \operatorname {Cauchy} (0,\gamma )}$ then ${\displaystyle {\tfrac {1}{X}}\sim \operatorname {Cauchy} (0,{\tfrac {1}{\gamma }})}$
• McCullagh's parametrization of the Cauchy distributions:^[28] Expressing a Cauchy distribution in terms of one complex parameter ${\displaystyle \psi =x_{0}+i\gamma }$, define ${\displaystyle X\sim \operatorname {Cauchy} (\psi )}$ to mean ${\displaystyle X\sim \operatorname {Cauchy} (x_{0},|\gamma |)}$. If ${\displaystyle X\sim \operatorname {Cauchy} (\psi )}$ then:
  $${\displaystyle {\frac {aX+b}{cX+d}}\sim \operatorname {Cauchy} \left({\frac {a\psi +b}{c\psi +d}}\right)}$$

  
  where ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ and ${\displaystyle d}$ are real numbers.
• Using the same convention as above, if ${\displaystyle X\sim \operatorname {Cauchy} (\psi )}$ then:^[28]
  $${\displaystyle {\frac {X-i}{X+i}}\sim \operatorname {CCauchy} \left({\frac {\psi -i}{\psi +i}}\right)}$$

  
  where ${\displaystyle \operatorname {CCauchy} }$ is the circular Cauchy distribution.

Lévy measure

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

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

where

${\displaystyle \Pi _{\gamma }(dy)=\left(c_{1,\gamma }{\frac {1}{y^{1+\gamma }}}1_{\left\{y>0\right\}}+c_{2,\gamma }{\frac {1}{|y|^{1+\gamma }}}1_{\left\{y<0\right\}}\right)\,dy}$

and ${\displaystyle c_{1,\gamma },c_{2,\gamma }}$ can be expressed explicitly.^[29] In the case ${\displaystyle \gamma =1}$ of the Cauchy distribution, one has ${\displaystyle c_{1,\gamma }=c_{2,\gamma }}$.

This last representation is a consequence of the formula

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

Related distributions

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

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.

Occurrence and applications

• 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.^[30] 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 ^[31] derived the test statistic for estimators of ${\displaystyle {\hat {\beta }}}$ for the equation ${\displaystyle x_{t+1}=\beta {x}_{t}+\varepsilon _{t+1},\beta >1}$ and where the maximum likelihood estimator is found using ordinary least squares showed the sampling distribution of the statistic is the Cauchy distribution.

Fitted cumulative Cauchy distribution to maximum one-day rainfalls using CumFreq, see also distribution fitting^[32]

• 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^[33] 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.^[34]

See also

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

References

External links

• "Cauchy distribution", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Earliest Uses: The entry on Cauchy distribution has some historical information.
• Weisstein, Eric W. "Cauchy Distribution". MathWorld.
• GNU Scientific Library – Reference Manual
• Ratios of Normal Variables by George Marsaglia