Log-Cauchy distribution

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Log-Cauchy
Probability density function
Log-Cauchy density function for values of [math]\displaystyle{ (\mu, \sigma) }[/math]
Cumulative distribution function
Log-Cauchy cumulative distribution function for values of [math]\displaystyle{ (\mu, \sigma) }[/math]
Parameters [math]\displaystyle{ \mu }[/math] (real)
[math]\displaystyle{ \displaystyle \sigma \gt 0\! }[/math] (real)
Support [math]\displaystyle{ \displaystyle x \in (0, +\infty)\! }[/math]
PDF [math]\displaystyle{ { 1 \over x\pi } \left[ { \sigma \over (\ln x - \mu)^2 + \sigma^2 } \right], \ \ x\gt 0 }[/math]
CDF [math]\displaystyle{ \frac{1}{\pi} \arctan\left(\frac{\ln x-\mu}{\sigma}\right)+\frac{1}{2}, \ \ x\gt 0 }[/math]
Mean infinite
Median [math]\displaystyle{ e^{\mu}\, }[/math]
Variance infinite
Skewness does not exist
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:

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

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

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

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

Cumulative distribution function

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

[math]\displaystyle{ F(x; 0, 1)=\frac{1}{2} + \frac{1}{\pi} \arctan(\ln x), \ \ x\gt 0 }[/math]

Survival function

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

[math]\displaystyle{ S(x; 0, 1)=\frac{1}{2} - \frac{1}{\pi} \arctan(\ln x), \ \ x\gt 0 }[/math]

Hazard rate

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

[math]\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\gt 0 }[/math]

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 [math]\displaystyle{ \mu }[/math].[1] The median absolute deviation of the natural logarithms of a sample is a robust estimator of [math]\displaystyle{ \sigma }[/math].[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. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 Olive, D.J. (June 23, 2008). "Applied Robust Statistics". Southern Illinois University. p. 86. http://www.math.siu.edu/olive/run.pdf. 
  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. https://www.proquest.com/openview/70d41b2007ad2f9ade89ed9ef6eba775/1?pq-origsite=gscholar&cbl=2035762. Retrieved 2022-04-01. 
  3. ↑ 3.0 3.1 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. https://archive.org/details/statisticalanaly00lind. 
  4. ↑ 4.0 4.1 4.2 4.3 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. https://archive.org/details/stochasticproces00mode. 
  5. ↑ 5.0 5.1 5.2 5.3 5.4 5.5 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. https://archive.org/details/lifedistribution00mars. 
  6. ↑ 6.0 6.1 Falk, M.; Hüsler, J.; Reiss, R. (2010). Laws of Small Numbers: Extremes and Rare Events. Springer. p. 80. ISBN 978-3-0348-0008-2. https://archive.org/details/lawssmallnumbers00falk. 
  7. ↑ Alves, M.I.F.; de Haan, L. (March 10, 2006). "Statistical inference for heavy and super-heavy tailed distributions". http://docentes.deio.fc.ul.pt/fragaalves/SuperHeavy.pdf. 
  8. ↑ "Moment". Mathworld. http://mathworld.wolfram.com/Moment.html. 
  9. ↑ Wang, Y.. Trade, Human Capital and Technology Spillovers: An Industry Level Analysis. Carleton University. p. 14. 
  10. ↑ Bondesson, L. (2003). "On the Lévy Measure of the Lognormal and LogCauchy Distributions". Methodology and Computing in Applied Probability: 243–256. http://resources.metapress.com/pdf-preview.axd?code=gn16hw202rxh4q1g&size=largest. Retrieved 2011-10-18. 
  11. ↑ Knight, J.; Satchell, S. (2001). Return distributions in finance. Butterworth-Heinemann. p. 153. ISBN 978-0-7506-4751-9. https://archive.org/details/returndistributi00satc_172. 
  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.. Statistical distributions in scientific work: proceedings of the NATO Advanced Study Institute. Springer. p. 169. ISBN 978-90-277-1334-6. 
  14. ↑ 14.0 14.1 Kleiber, C.; Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Science. Wiley. pp. 101–102, 110. ISBN 978-0-471-15064-0. https://archive.org/details/statisticalsized0000unse. 
  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. 
  19. ↑ Zuo-Yun, Y. (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. 



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