Nonparametric estimation of Matusita's measure of affinity between absolutely continuous distributions
 Ibrahim A. Ahmad
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LetF andG be two distribution functions defined on the same probability space which are absolutely continuous with respect to the Lebesgue measure with probability densitiesf andg, respectively. Matusita [3] defines a measure of the closeness, affinity, betweenF andG as: \(\rho = \rho (F,G) = \int {[f(x)g(x)]^{1/2} } dx\) . Based on two independent samples fromF andG we propose to estimate ρ by \(\hat \rho = \int {[\hat f(x)\hat g(x)]^{1/2} } dx\) , where \(\hat f(x)\) and \(\hat g(x)\) are taken to be the kernel estimates off(x) andg(x), respectively, as given by Parzen [5].
In this note sufficient conditions are given such that (i) \(E(\hat \rho  \rho )^2 \to 0\) asx→∞ and (ii) \(\hat \rho  \rho \) with probability one, asn→∞.
 Ahmad, I. A. and Van Belle, G. (1974). Measuring affinity of distributions.Reliability and Biometry, Statistical Analysis of Life Testing, (eds. F. Proschan and R. J. Serfling), SIAM, Philadelphia, 651–668.
 Ahmad, I. A. (1980). Nonparametric estimation of an affinity measure between two absolutely continuous distributions with hypothesis testing applications,Ann. Inst. Statist. Math.,32, 223–240.
 Matusita, K. (1955). Decision rules based on the distance for the problem of fit, two samples and estimation,Ann. Math. Statist.,26, 631–640.
 Matusita, K. (1967). On the notion of affinity of several distributions and some of its applications,Ann. Inst. Statist. Math.,19, 181–192. CrossRef
 Parzen, E. (1962). On estimation of a probability density function and mode,Ann. Math. Statist.,33, 1065–1076.
 Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function,Ann. Math. Statist.,27, 832–837.
 Royden, H. L. (1968).Real Analysis, (Second Edition), Macmillan, New York.
 Title
 Nonparametric estimation of Matusita's measure of affinity between absolutely continuous distributions
 Journal

Annals of the Institute of Statistical Mathematics
Volume 32, Issue 1 , pp 241245
 Cover Date
 19801201
 DOI
 10.1007/BF02480328
 Print ISSN
 00203157
 Online ISSN
 15729052
 Publisher
 Kluwer Academic Publishers
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 Authors

 Ibrahim A. Ahmad ^{(1)}
 Author Affiliations

 1. MacMaster University, Canada