On the convexity of higher order Jensen differences based on entropy functions

Abstract

In an earlier work, the authors introduced a divergence measure, called the first-order Jensen difference, or in shortcal j-divergence, which is based on entropy functions of degreealpha. This provided a generalization of the measure of mutual information based on Shannon's entropy (corresponding to β = 1). It was shown that the first-ordercal j-divergence is a convex function only when a is restricted to some range. We define higher order Jensen differences and show that they are convex functions only when the underlying entropy function is of degree two. A statistical application requiring the convexity of higher order Jensen differences is indicated.