Gram–Charlier-like expansions of power-raised hyperbolic secant laws

Spherical distributions arise quite naturally as multivariate versions of univariate (even) densities and prove useful in several applications. Likewise their univariate counterparts, they may not always meet the kurtosis requirements of empirical evidence. This paper devises a methodological approach which duly reshapes spherical distributions to match kurtosis requirements to due extent. This approach is tailored to the family of power-raised hyperbolic secant laws and hinges on Gram–Charlier-like expansions via second-degree orthogonal polynomials.