Log-Cauchy distribution explained

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.

Characterization

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

Probability density function

The log-Cauchy distribution has the probability density function:

\begin{align} f(x; \mu,\sigma) & = (1)/(x\pi\sigma \left[ 1 + \left( (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{align}

where

\mu is a real number and \sigma > 0 .^{[2] [3]} If \sigma is known, the scale parameter is e ^{\mu} .^[2] \mu and \sigma correspond to the location parameter and scale parameter of the associated Cauchy distribution.^{[2] [4]} Some authors define \mu and \sigma as the location and scale parameters, respectively, of the log-Cauchy distribution.^[4]

For

\mu = 0 and \sigma =1 , corresponding to a standard Cauchy distribution, the probability density function reduces to:f(x; 0 , 1 ) = (1)/(x\pi [ 1+ ( ln x) ^{2]}) , x> 0

Cumulative distribution function

The cumulative distribution function (cdf) when

\mu = 0 and \sigma =1 is:F(x; 0 , 1 ) = (1)/(2) + (1)/(\pi) \arctan( ln x), x> 0

Survival function

The survival function when

\mu = 0 and \sigma =1 is:S(x; 0 , 1 ) = (1)/(2) - (1)/(\pi) \arctan( ln x), x> 0

Hazard rate

The hazard rate when

\mu = 0 and \sigma =1 is:λ (x; 0 , 1 ) = \left\{ (1)/(\left[ x\pi \left[ 1+ \left( ln x\right) ^{2\right]}) (1)/(2) - (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.

Properties

The log-Cauchy distribution is an example of a heavy-tailed distribution.^[5] 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.^{[5] [6]} As with the Cauchy distribution, none of the non-trivial moments of the log-Cauchy distribution are finite.^[7] 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

\mu .^[2] The median absolute deviation of the natural logarithms of a sample is a robust estimator of \sigma .^[2]

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]

Notes and References

1. Revista Colombiana de Estadística - Applied Statistics. 2022-04-01. January 2022. 45. 1. 209–229. Some Inferential Problems from Log Student's T-distribution and its Multivariate Extension. Olosunde, Akinlolu & Olofintuade, Sylvester. 10.15446/rce.v45n1.90672. free.
2. Web site: Applied Robust Statistics. Olive, D.J.. June 23, 2008. Southern Illinois University. 86. 2011-10-18. dead. https://web.archive.org/web/20110928191222/http://www.math.siu.edu/olive/run.pdf. September 28, 2011.
3. Book: Statistical analysis of stochastic processes in time. limited. Lindsey, J.K.. 33, 50, 56, 62, 145. 2004. Cambridge University Press. 978-0-521-83741-5.
4. Book: Stochastic processes in epidemiology: HIV/AIDS, other infectious diseases. limited. Mode, C.J. . Sleeman, C.K. . amp . 29–37. 2000. World Scientific. 978-981-02-4097-4.
5. Book: Laws of Small Numbers: Extremes and Rare Events. limited. Falk, M.. Hüsler, J.. Reiss, R.. amp. 80. 2010. Springer. 978-3-0348-0008-2.
6. Web site: Statistical inference for heavy and super-heavy tailed distributions . https://web.archive.org/web/20070623175435/http://docentes.deio.fc.ul.pt/fragaalves/SuperHeavy.pdf . dead . June 23, 2007 . Alves, M.I.F. . de Haan, L. . Neves, C. . amp . March 10, 2006 .
7. Book: Life distributions: structure of nonparametric, semiparametric, and parametric families. limited. Marshall, A.W. . Olkin, I. . amp . 443–444. 2007. Springer. 978-0-387-20333-1.
8. Web site: Moment. Mathworld. 2011-10-19.
9. Trade, Human Capital and Technology Spillovers: An Industry Level Analysis. Wang, Y.. 14. Carleton University.
10. On the Lévy Measure of the Lognormal and LogCauchy Distributions. 2011-10-18. Bondesson, L.. Methodology and Computing in Applied Probability. 2003. 243–256. dead. https://web.archive.org/web/20120425064706/http://resources.metapress.com/pdf-preview.axd?code=gn16hw202rxh4q1g&size=largest. 2012-04-25.
11. Book: Return distributions in finance. limited. Knight, J. . Satchell, S. . amp . 153. 2001. Butterworth-Heinemann. 978-0-7506-4751-9.
12. Book: Market consistency: model calibration in imperfect markets. Kemp, M.. 2009. Wiley. 978-0-470-77088-7.
13. Book: Statistical distributions in scientific work: proceedings of the NATO Advanced Study Institute. MacDonald, J.B.. Measuring Income Inequality. 169. Taillie, C. . Patil, G.P. . Baldessari, B.. 1981. Springer. 978-90-277-1334-6.
14. Book: Statistical Size Distributions in Economics and Actuarial Science. registration. Kleiber, C. . Kotz, S. . amp . 101–102, 110. 2003. Wiley. 978-0-471-15064-0.
15. Distribution function values for logstable distributions. 10.1016/0898-1221(93)90128-I. Panton, D.B.. May 1993. 17–24. 25. 9. Computers & Mathematics with Applications. free.
16. Book: Good thinking: the foundations of probability and its applications. Good, I.J.. 102. 1983. University of Minnesota Press. 978-0-8166-1142-3.
17. Book: Frontiers of Statistical Decision Making and Bayesian Analysis. 12. Chen, M.. 2010. Springer. 978-1-4419-6943-9.
18. Some statistical issues in modelling pharmacokinetic data. Lindsey, J.K.. Jones, B.. Jarvis, P.. amp. Statistics in Medicine. September 2001. 20. 17–18. 2775–278. 10.1002/sim.742. 11523082. 41887351 .
19. LogCauchy, log-sech and lognormal distributions of species abundances in forest communities. Ecological Modelling. 184. 2–4. 10.1016/j.ecolmodel.2004.10.011. June 2005. 329–340. Zuo-Yun, Y.. etal .