Log-Cauchy distribution

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
Ex. 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

1. Olive, D.J. (June 23, 2008). "Applied Robust Statistics" (PDF). Southern Illinois University. p. 86. Archived from the original (PDF) on September 28, 2011. Retrieved 2011-10-18.
2. Olosunde, Akinlolu & Olofintuade, Sylvester (January 2022). "Some Inferential Problems from Log Student's T-distribution and its Multivariate Extension". Revista Colombiana de Estadística - Applied Statistics. 45 (1): 209–229. doi: 10.15446/rce.v45n1.90672 . Retrieved 2022-04-01.
3. Lindsey, J.K. (2004). Statistical analysis of stochastic processes in time . Cambridge University Press. pp.  33, 50, 56, 62, 145. ISBN   978-0-521-83741-5.
4. Mode, C.J. & Sleeman, C.K. (2000). Stochastic processes in epidemiology: HIV/AIDS, other infectious diseases . World Scientific. pp.  29–37. ISBN   978-981-02-4097-4.
5. Marshall, A.W. & Olkin, I. (2007). Life distributions: structure of nonparametric, semiparametric, and parametric families . Springer. pp.  443–444. ISBN   978-0-387-20333-1.
6. Falk, M.; Hüsler, J. & Reiss, R. (2010). Laws of Small Numbers: Extremes and Rare Events . Springer. p.  80. ISBN   978-3-0348-0008-2.
7. Alves, M.I.F.; de Haan, L. & Neves, C. (March 10, 2006). "Statistical inference for heavy and super-heavy tailed distributions" (PDF). Archived from the original (PDF) on June 23, 2007.
8. "Moment". Mathworld . Retrieved 2011-10-19.
9. Wang, Y. "Trade, Human Capital and Technology Spillovers: An Industry Level Analysis". Carleton University: 14.
10. Bondesson, L. (2003). "On the Lévy Measure of the Lognormal and LogCauchy Distributions". Methodology and Computing in Applied Probability: 243–256. Archived from the original on 2012-04-25. Retrieved 2011-10-18.
11. Knight, J. & Satchell, S. (2001). Return distributions in finance . Butterworth-Heinemann. p.  153. ISBN   978-0-7506-4751-9.
12. Kemp, M. (2009). Market consistency: model calibration in imperfect markets. Wiley. ISBN   978-0-470-77088-7.
13. MacDonald, J.B. (1981). "Measuring Income Inequality". In Taillie, C.; Patil, G.P.; Baldessari, B. (eds.). Statistical distributions in scientific work: proceedings of the NATO Advanced Study Institute. Springer. p. 169. ISBN   978-90-277-1334-6.
14. Kleiber, C. & Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Science . Wiley. pp.  101–102, 110. ISBN   978-0-471-15064-0.
15. Panton, D.B. (May 1993). "Distribution function values for logstable distributions". Computers & Mathematics with Applications. 25 (9): 17–24. doi: 10.1016/0898-1221(93)90128-I .
16. Good, I.J. (1983). Good thinking: the foundations of probability and its applications. University of Minnesota Press. p. 102. ISBN   978-0-8166-1142-3.
17. Chen, M. (2010). Frontiers of Statistical Decision Making and Bayesian Analysis. Springer. p. 12. ISBN   978-1-4419-6943-9.
18. Lindsey, J.K.; Jones, B. & Jarvis, P. (September 2001). "Some statistical issues in modelling pharmacokinetic data". Statistics in Medicine. 20 (17–18): 2775–278. doi:10.1002/sim.742. PMID   11523082. S2CID   41887351.
19. Zuo-Yun, Y.; et al. (June 2005). "LogCauchy, log-sech and lognormal distributions of species abundances in forest communities". Ecological Modelling. 184 (2–4): 329–340. doi:10.1016/j.ecolmodel.2004.10.011.