Diskussionspapiere // Friedrich-Alexander-Universität Erlangen-Nürnberg, Lehrstuhl für Statistik und Ökonometrie 64/2004
The shape of a probability distribution is often summarized by the distribution's skewness and kurtosis. Starting from a symmetric parent density f on the real line, we can modify its shape (i.e. introduce skewness and in-/decrease kurtosis) if f is appropriately weighted. In particular, every density w on the interval (0; 1) is a specific weighting function. Within this work, we follow up a proposal of Jones (2004) and choose the Beta distribution as underlying weighting function w. Parent distributions like the Student-t, the logistic and the normal distribution have already been investigated in the literature. Based on the assumption that f is the density of a hyperbolic secant distribution, we introduce the Beta-hyperbolic secant (BHS) distribution. In contrast to the Beta-normal distribution and the to Beta-Student-t distribution, BHS densities are always unimodal and all moments exist. In contrast to the Beta-logistic distribution, the BHS distribution is more êexible regarding the range of skewness and leptokurtosis combinations. Moreover, we propose a generalization which nests both the Beta-logistic and the BHS distribution. Finally, the goodness-of-fit between all above-mentioned distributions is compared for glass fibre data and aluminium returns.