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

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

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

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

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.