Annals of the Institute of Statistical Mathematics

, Volume 32, Issue 1, pp 223–240

Nonparametric estimation of an affinity measure between two absolutely continuous distributions with hypotheses testing applications

  • Ibrahim A. Ahmad

DOI: 10.1007/BF02480327

Cite this article as:
Ahmad, I.A. Ann Inst Stat Math (1980) 32: 223. doi:10.1007/BF02480327


LetF andG denote two distribution functions defined on the same probability space and are absolutely continuous with respect to the Lebesgue measure with probability density functionsf andg, respectively. A measure of the closeness betweenF andG is defined by:\(\lambda = \lambda (F,G) = 2\int {f(x)g(x)dx} /\left[ {\int {f^2 (x)dx + \int {g^2 (x)dx} } } \right]\). Based on two independent samples it is proposed to estimate λ by\(\hat \lambda = \left[ {\int {\hat f(x)dG_n (x) + \int {\hat g(x)dF_n (x)} } } \right]/\left[ {\int {\hat f^2 (x)dx + \int {\hat g^2 (x)dx} } } \right]\), whereFn(x) andGn(x) are the empirical distribution functions ofF(x) andG(x) respectively and\(\hat f(x)\) and\(\hat g(x)\) are taken to be the so-called kernel estimates off(x) andg(x) respectively, as defined by Parzen [16]. Large sample theory of\(\hat \lambda \) is presented and a two sample goodness-of-fit test is presented based on\(\hat \lambda \). Also discussed are estimates of certain modifications of λ which allow us to propose some test statistics for the one sample case, i.e., wheng(x)=f0(x), withf0(x) completely known and for testing symmetry, i.e., testingH0:f(x)=f(−x).

Copyright information

© Kluwer Academic Publishers 1980

Authors and Affiliations

  • Ibrahim A. Ahmad
    • 2
  1. 1.Department of Mathematical SciencesMemphis State UniversityMemphisU.S.A.
  2. 2.McMaster UniversityCanada

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