Annals of the Institute of Statistical Mathematics

, Volume 32, Issue 1, pp 241–245

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

• Ibrahim A. Ahmad
Article

## 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→∞.

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### References

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