On the convexity of higher order Jensen differences based on entropy functions (Corresp.)

In an earlier work, the authors introduced a divergence measure, called the first-order Jensen difference, or in short
\cal j
-divergence, which is based on entropy functions of degree
\alpha
. This provided a generalization of the measure of mutual information based on Shannon's entropy (corresponding to
\alpha = 1)
. It was shown that the first-order
\cal 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.