Log-Cauchy

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \mu }$ ( real) ${\displaystyle \displaystyle \sigma >0\!}$ (real)
Support	${\displaystyle \displaystyle x\in (0,+\infty )\!}$
PDF	${\displaystyle {1 \over x\pi }\left[{\sigma \over (\ln x-\mu )^{2}+\sigma ^{2}}\right],\ \ x>0}$
CDF	${\displaystyle {\frac {1}{\pi }}\arctan \left({\frac {\ln x-\mu }{\sigma }}\right)+{\frac {1}{2}},\ \ x>0}$
Mean	infinite
Median	${\displaystyle e^{\mu }\,}$
Variance	infinite
Skewness	does not exist
Excess kurtosis	does not exist
MGF	does not exist

In probability theory, a log-Cauchy distribution is a probability distribution of a random variable whose logarithm is distributed in accordance with a Cauchy distribution. If X is a random variable with a Cauchy distribution, then Y = exp(X) has a log-Cauchy distribution; likewise, if Y has a log-Cauchy distribution, then X = log(Y) has a Cauchy distribution. ^[1]

Characterization

The log-Cauchy distribution is a special case of the log-t distribution where the degrees of freedom parameter is equal to 1. ^[2]

Probability density function

The log-Cauchy distribution has the probability density function:

${\displaystyle {\begin{aligned}f(x;\mu ,\sigma )&={\frac {1}{x\pi \sigma \left[1+\left({\frac {\ln x-\mu }{\sigma }}\right)^{2}\right]}},\ \ x>0\\&={1 \over x\pi }\left[{\sigma \over (\ln x-\mu )^{2}+\sigma ^{2}}\right],\ \ x>0\end{aligned}}}$

where ${\displaystyle \mu }$ is a real number and ${\displaystyle \sigma >0}$. ^{[1] [3]} If ${\displaystyle \sigma }$ is known, the scale parameter is ${\displaystyle e^{\mu }}$. ^[1] ${\displaystyle \mu }$ and ${\displaystyle \sigma }$ correspond to the location parameter and scale parameter of the associated Cauchy distribution. ^{[1] [4]} Some authors define ${\displaystyle \mu }$ and ${\displaystyle \sigma }$ as the location and scale parameters, respectively, of the log-Cauchy distribution. ^[4]

For ${\displaystyle \mu =0}$ and ${\displaystyle \sigma =1}$, corresponding to a standard Cauchy distribution, the probability density function reduces to: ^[5]

${\displaystyle f(x;0,1)={\frac {1}{x\pi [1+(\ln x)^{2}]}},\ \ x>0}$

Cumulative distribution function

The cumulative distribution function ( cdf) when ${\displaystyle \mu =0}$ and ${\displaystyle \sigma =1}$ is: ^[5]

${\displaystyle F(x;0,1)={\frac {1}{2}}+{\frac {1}{\pi }}\arctan(\ln x),\ \ x>0}$

Survival function

The survival function when ${\displaystyle \mu =0}$ and ${\displaystyle \sigma =1}$ is: ^[5]

${\displaystyle S(x;0,1)={\frac {1}{2}}-{\frac {1}{\pi }}\arctan(\ln x),\ \ x>0}$

Hazard rate

The hazard rate when ${\displaystyle \mu =0}$ and ${\displaystyle \sigma =1}$ is: ^[5]

${\displaystyle \lambda (x;0,1)=\left\{{\frac {1}{x\pi \left[1+\left(\ln x\right)^{2}\right]}}\left[{\frac {1}{2}}-{\frac {1}{\pi }}\arctan(\ln x)\right]\right\}^{-1},\ \ x>0}$

The hazard rate decreases at the beginning and at the end of the distribution, but there may be an interval over which the hazard rate increases. ^[5]

Properties

The log-Cauchy distribution is an example of a heavy-tailed distribution. ^[6] Some authors regard it as a "super-heavy tailed" distribution, because it has a heavier tail than a Pareto distribution-type heavy tail, i.e., it has a logarithmically decaying tail. ^{[6] [7]} As with the Cauchy distribution, none of the non-trivial moments of the log-Cauchy distribution are finite. ^[5] The mean is a moment so the log-Cauchy distribution does not have a defined mean or standard deviation. ^{[8] [9]}

The log-Cauchy distribution is infinitely divisible for some parameters but not for others. ^[10] Like the lognormal distribution, log-t or log-Student distribution and Weibull distribution, the log-Cauchy distribution is a special case of the generalized beta distribution of the second kind. ^{[11] [12]} The log-Cauchy is actually a special case of the log-t distribution, similar to the Cauchy distribution being a special case of the Student's t distribution with 1 degree of freedom. ^{[13] [14]}

Since the Cauchy distribution is a stable distribution, the log-Cauchy distribution is a logstable distribution. ^[15] Logstable distributions have poles at x=0. ^[14]

Estimating parameters

The median of the natural logarithms of a sample is a robust estimator of ${\displaystyle \mu }$. ^[1] The median absolute deviation of the natural logarithms of a sample is a robust estimator of ${\displaystyle \sigma }$. ^[1]

Uses

In Bayesian statistics, the log-Cauchy distribution can be used to approximate the improper Jeffreys-Haldane density, 1/k, which is sometimes suggested as the prior distribution for k where k is a positive parameter being estimated. ^{[16] [17]} The log-Cauchy distribution can be used to model certain survival processes where significant outliers or extreme results may occur. ^{[3] [4] [18]} An example of a process where a log-Cauchy distribution may be an appropriate model is the time between someone becoming infected with HIV and showing symptoms of the disease, which may be very long for some people. ^[4] It has also been proposed as a model for species abundance patterns. ^[19]

References