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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→∞.

Keywords

Probability Density Function Lebesgue Measure Probability Space Continuous Distribution Nonparametric Estimation 

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References

  1. [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. [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. [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. [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. [5]
    Parzen, E. (1962). On estimation of a probability density function and mode,Ann. Math. Statist.,33, 1065–1076.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function,Ann. Math. Statist.,27, 832–837.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Royden, H. L. (1968).Real Analysis, (Second Edition), Macmillan, New York.zbMATHGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 1980

Authors and Affiliations

  • Ibrahim A. Ahmad
    • 1
  1. 1.MacMaster UniversityCanada

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