RELATIVE DIFFERENCE IN DIVERSITY BETWEEN POPULATIONS

KHURSHEED ALAM AND CALVIN L. WILLIAMS

Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-1907, U.S.A.

(Received June 15, 1991; revised May 26, 1992)

Abstract. An entropy is conceived as a functional on the space of probability distributions. It is used as a measure of diversity (variability) of a population. Cross entropy leads to a measure of dissimilarity between populations. In this paper, we provide a new approach to the construction of a measure of dissimilarity between two populations, not depending on the choice of an entropy function, measuring diversity. The approach is based on the principle of majorization which provides an intrinsic method of comparing the diversities of two populations. We obtain a general class of measures of dissimilarity and show some interesting properties of the proposed index. In particular, it is shown that the measure provides a metric on a probability space. The proposed measure of dissimilarity is essentially a measure of relative difference in diversity between two populations. It satisfies an invariance property which is not shared by other measures of dissimilarity which are used in ecological studies. A statistical application of the new method is given.

Key words and phrases: Diversity, dissimilarity, cross entropy, majorization, Schur-convexity, ranking and selection.