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

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DOI: 10.1007/BF02480328

- Cite this article as:
- Ahmad, I.A. Ann Inst Stat Math (1980) 32: 241. doi:10.1007/BF02480328

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## Abstract

Let*F* and*G* be two distribution functions defined on the same probability space which are absolutely continuous with respect to the Lebesgue measure with probability densities*f* and*g*, respectively. Matusita [3] defines a measure of the closeness, affinity, between*F* and*G* as:\(\rho = \rho (F,G) = \int {[f(x)g(x)]^{1/2} } dx\). Based on two independent samples from*F* and*G* 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 of*f(x)* and*g(x)*, respectively, as given by Parzen [5].

In this note sufficient conditions are given such that (i)\(E(\hat \rho - \rho )^2 \to 0\) as*x*→∞ and (ii)\(\hat \rho - \rho \) with probability one, as*n*→∞.