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

Abstract

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→∞.