Log-Laplace distribution

In probability theory and statistics, the log-Laplace distribution is the probability distribution of a random variable whose logarithm has a Laplace distribution. If X has a Laplace distribution with parameters μ and b, then Y = e^X has a log-Laplace distribution. The distributional properties can be derived from the Laplace distribution.

Characterization

Probability density function

A random variable has a log-Laplace(μ, b) distribution if its probability density function is:^[1]

$f(x|\mu ,b)={\frac {1}{2bx}}\exp \left(-{\frac {|\ln x-\mu |}{b}}\right)\,\!$

$={\frac {1}{2bx}}\left\{{\begin{matrix}\exp \left(-{\frac {\mu -\ln x}{b}}\right)&{\mbox{if }}x<\mu \\[8pt]\exp \left(-{\frac {\ln x-\mu }{b}}\right)&{\mbox{if }}x\geq \mu \end{matrix}}\right.$

The cumulative distribution function for Y when y > 0, is

${\displaystyle F(y)=0.5\,[1+\operatorname {sgn}(\ln(y)-\mu )\,(1-\exp(-|\ln(y)-\mu |/b))].}$

Versions of the log-Laplace distribution based on an asymmetric Laplace distribution also exist.^[2] Depending on the parameters, including asymmetry, the log-Laplace may or may not have a finite mean and a finite variance.^[2]

Differential equation

$\left\{{\begin{matrix}\left\{bxf'(x)+(b-1)f(x)=0,f(1)={\frac {e^{{-{\frac {\mu }{b}}}}}{2b}}\right\}&{\mbox{if }}x<\mu \\[8pt]\left\{bxf'(x)+(b+1)f(x)=0,f(1)={\frac {e^{{{\frac {\mu }{b}}}}}{2b}}\right\}&{\mbox{if }}x\geq \mu \end{matrix}}\right.$

References

1. Lindsey, J.K. (2004). Statistical analysis of stochastic processes in time. Cambridge University Press. p. 33. ISBN 978-0-521-83741-5.
2. Kozubowski, T.J. & Podgorski, K. "A Log-Laplace Growth Rate Model" (PDF). University of Nevada-Reno. p. 4. Retrieved 2011-10-21.