On the skew Laplace distribution

A random variable X is said to have the skew-Laplace distribution if its pdf is f(x) = 2g(x)G(λx), where g(·) and G(·), respectively, denote the pdf and the cdf of the Laplace distribution. This distribution – in spite of its simplicity – appears not to have been studied in detail. The only work that appears to give some details of this distribution is Gupta et. al [Random Operators and Stochastic Equations, Vol. 10 (2002), pp. 133–140], where expressions for the expectation, variance, skewness and the kurtosis of X are given. But these expressions appear to contain some errors. In this paper, we provide a comprehensive description of the mathematical properties of X. The properties derived include the k th moment, variance, skewness, kurtosis, moment generating function, characteristic function, cumulant generating function, the k th cumulant, hazard rate function, mean deviation about the mean, mean deviation about the median, Renyi entropy, Shannon's entropy, cumulative residual entropy and the asymptotic distribution of the extreme order statistics. We also consider estimation and simulation issues.