Log-Cauchy distribution

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

Probability density function

The log-Cauchy distribution has the probability density function:

{\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 $\mu$ is a real number and $\sigma >0$ .^[1]^[2] If $\sigma$ is known, the scale parameter is $e^{{\mu }}$ .^[1] $\mu$ and $\sigma$ correspond to the location parameter and scale parameter of the associated Cauchy distribution.^[1]^[3] Some authors define $\mu$ and $\sigma$ as the location and scale parameters, respectively, of the log-Cauchy distribution.^[3]

For $\mu =0$ and $\sigma =1$ , corresponding to a standard Cauchy distribution, the probability density function reduces to:^[4]

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

Cumulative distribution function

The cumulative distribution function (cdf) when $\mu =0$ and $\sigma =1$ is:^[4]

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

Survival function

The survival function when $\mu =0$ and $\sigma =1$ is:^[4]

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

Hazard rate

The hazard rate when $\mu =0$ and $\sigma =1$ is:^[4]

\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.^[4]

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.^[4] The mean is a moment so the log-Cauchy distribution does not have a defined mean or standard deviation.^[7]^[8]

The log-Cauchy distribution is infinitely divisible for some parameters but not for others.^[9] 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.^[10]^[11] 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.^[12]^[13]

Since the Cauchy distribution is a stable distribution, the log-Cauchy distribution is a logstable distribution.^[14] Logstable distributions have poles at x=0.^[13]

Estimating parameters

The median of the natural logarithms of a sample is a robust estimator of $\mu$ .^[1] The median absolute deviation of the natural logarithms of a sample is a robust estimator of $\sigma$ .^[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.^[15]^[16] The log-Cauchy distribution can be used to model certain survival processes where significant outliers or extreme results may occur.^[2]^[3]^[17] An example of a process where a log-Cauchy distribution may be an appropriate model is the time between someone becoming infected with HIV virus and showing symptoms of the disease, which may be very long for some people.^[3] It has also been proposed as a model for species abundance patterns.^[18]

References

1 2 3 4 5 6 Olive, D.J. (June 23, 2008). "Applied Robust Statistics" (PDF). Southern Illinois University. p. 86. Retrieved 2011-10-18.
1 2 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.
1 2 3 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.
1 2 3 4 5 6 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.
1 2 Falk, M., Hüsler, J. & Reiss, R. (2010). Laws of Small Numbers: Extremes and Rare Events. Springer. p. 80. ISBN 978-3-0348-0008-2.
↑ Alves, M.I.F., de Haan, L. & Neves, C. (March 10, 2006). "Statistical inference for heavy and super-heavy tailed distributions" (PDF).
↑ "Moment". Mathworld. Retrieved 2011-10-19.
↑ Wang, Y. "Trade, Human Capital and Technology Spillovers: An Industry Level Analysis". Carleton University: 14.
↑ Bondesson, L. (2003). "On the Lévy Measure of the Lognormal and LogCauchy Distributions". Methodology and Computing in Applied Probability. Kluwer Academic Publications: 243–256. Retrieved 2011-10-18.
↑ Knight, J. & Satchell, S. (2001). Return distributions in finance. Butterworth-Heinemann. p. 153. ISBN 978-0-7506-4751-9.
↑ Kemp, M. (2009). Market consistency: model calibration in imperfect markets. Wiley. ISBN 978-0-470-77088-7.
↑ MacDonald, J.B. (1981). "Measuring Income Inequality". In Taillie, C.; Patil, G.P.; Baldessari, B. Statistical distributions in scientific work: proceedings of the NATO Advanced Study Institute. Springer. p. 169. ISBN 978-90-277-1334-6.
1 2 Kleiber, C. & Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Science. Wiley. pp. 101–102, 110. ISBN 978-0-471-15064-0.
↑ 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. Retrieved 2011-10-18.
↑ 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.
↑ Chen, M. (2010). Frontiers of Statistical Decision Making and Bayesian Analysis. Springer. p. 12. ISBN 978-1-4419-6943-9.
↑ Lindsey, J.K., Jones, B. & Jarvis, P.; Jones; Jarvis (September 2001). "Some statistical issues in modelling pharmacokinetic data". Statistics in Medicine. 20 (17–18): 2775–278. doi:10.1002/sim.742. Retrieved 2011-10-19.
↑ 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. Retrieved 2011-10-18.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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