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Convexity Properties of Entropy Functions and Analysis of Diversity

C. Radhakrishna Rao
Lecture Notes-Monograph Series
Vol. 5, Inequalities in Statistics and Probability (1984), pp. 68-77
Stable URL: http://www.jstor.org/stable/4355484
Page Count: 10
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Convexity Properties of Entropy Functions and Analysis of Diversity
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Abstract

Some natural conditions which a diversity measure (variability) of a probability distribution should satisfy imply that it must have certain convexity properties, considered as a functional on the space of probability distributions. It is shown that some of the well known entropy functions, which are used as diversity measures do not have all the desirable properties and are, therefore, of limited use. A new measure called the quadratic entropy has been introduced, which seems to be well suited for studying diversity. Methods for apportioning diversity (APDIV) at various levels of a hierarchically classified set of populations are described. The concept of analysis of diversity (ANODIV), as a generalization of ANOVA, applicable to observations of any type, is developed and its use in the analysis of cross classified data is demonstrated. The choice of a suitable measure of diversity for the above purpose is discussed.

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