Peakedness and Generalized Entropy for Continuous Density Functions

Abstract

The theory of majorisation between real vectors with equal sum of components, originated in the beginning of the XXth century, enables a partial ordering between discrete probability distributions to be defined. It corresponds to comparing, via fuzzy set inclusion, possibility distributions that are the most specific transforms of the original probability distributions. This partial ordering compares discrete probability distributions in terms of relative peakedness around their mode, and entropy is monotonic with respect to this partial ordering. In fact, all known variants of entropy share this monotonicity. In this paper, this question is studied in the case of unimodal continuous probability densities on the real line, for which a possibility transform around the mode exists. It corresponds to extracting the family of most precise prediction intervals. Comparing such prediction intervals for two densities yields a variant of relative peakedness in the sense of Birnbaum. We show that a generalized form of continuous entropy is monotonic with respect to this form of relative peakedness of densities.