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RELATIVE DIFFERENCE IN DIVERSITY BETWEEN POPULATIONS

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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.

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