Kurtosis ordering of the generalized secant hyperbolic distribution: a technical note

Two major generalizations of the hyperbolic secant distribution have been proposed in the statistical literature which both introduce an additional parameter that governs the kurtosis of the generalized distribution. The generalized hyperbolic secant (GHS) distribution was introduced by Harkness and Harkness (1968) who considered the p-th convolution of a hyperbolic secant distribution. Another generalization, the so-called generalized secant hyperbolic (GSH) distribution was recently suggested by Vaughan 2002). In contrast to the GHS distribution, the cumulative and inverse cumulative distribution function of the GSH distribution are available in closedform expressions. We use this property to proof that the additional shape parameter of the GSH distribution is actually a kurtosis parameter in the sense of van Zwet (1964).