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


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→∞.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


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

Copyright information

© Kluwer Academic Publishers 1980

Authors and Affiliations

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

Personalised recommendations