Annals of the Institute of Statistical Mathematics

, Volume 32, Issue 1, pp 241-245

First online:

Nonparametric estimation of Matusita's measure of affinity between absolutely continuous distributions

  • Ibrahim A. AhmadAffiliated withMacMaster University

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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→∞.