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

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