The skew logistic distribution

A random variable X is said to have Azzalini’s skew-logistic distribution if its pdf is f(x)=2g(x)G(λ x), where g(⋅) and G(⋅), respectively, denote the pdf and cdf of the logistic distribution. This distribution—in spite of its simplicity—appears not to have been studied in detail. In this note, we provide a comprehensive description of the mathematical properties of X. The properties derived include the cumulative distribution function, the nth moment (including expressions for the first ten moments), the nth central moment, moment generating function, characteristic function, mean deviation about the mean, mean deviation about the median, Rényi entropy, Shannon entropy, order statistics, the asymptotic distribution of the extreme order statistics, estimation by the methods of moments and maximum likelihood, the associated Fisher information matrix and simulation issues. An application to logistic regression is discussed.