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

- 22 Downloads
- 6 Citations

## 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*→∞.

## Keywords

Probability Density Function Lebesgue Measure Probability Space Continuous Distribution Nonparametric Estimation## Preview

Unable to display preview. Download preview PDF.

## References

- [1]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.Google Scholar - [2]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.MathSciNetCrossRefGoogle Scholar - [3]Matusita, K. (1955). Decision rules based on the distance for the problem of fit, two samples and estimation,
*Ann. Math. Statist.*,**26**, 631–640.MathSciNetCrossRefGoogle Scholar - [4]Matusita, K. (1967). On the notion of affinity of several distributions and some of its applications,
*Ann. Inst. Statist. Math.*,**19**, 181–192.MathSciNetCrossRefGoogle Scholar - [5]Parzen, E. (1962). On estimation of a probability density function and mode,
*Ann. Math. Statist.*,**33**, 1065–1076.MathSciNetCrossRefGoogle Scholar - [6]Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function,
*Ann. Math. Statist.*,**27**, 832–837.MathSciNetCrossRefGoogle Scholar - [7]Royden, H. L. (1968).
*Real Analysis*, (Second Edition), Macmillan, New York.zbMATHGoogle Scholar