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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> j</tex> -divergence, which is based on entropy functions of degree <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</tex> . This provided a generalization of the measure of mutual information based on Shannon's entropy (corresponding to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α = 1)</tex> . It was shown that the first-order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> j</tex> -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.