Nonparametric estimation of an affinity measure between two absolutely continuous distributions with hypotheses testing applications
 Ibrahim A. Ahmad
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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 \(\hat \lambda = \left[ {\int {\hat f(x)dG_n (x) + \int {\hat g(x)dF_n (x)} } } \right]/\left[ {\int {\hat f^2 (x)dx + \int {\hat g^2 (x)dx} } } \right]\) , whereF _{ n } (x) andG _{ n } (x) are the empirical distribution functions ofF(x) andG(x) respectively and \(\hat f(x)\) and \(\hat g(x)\) are taken to be the socalled kernel estimates off(x) andg(x) respectively, as defined by Parzen [16]. Large sample theory of \(\hat \lambda \) is presented and a two sample goodnessoffit 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)=f _{0} (x), withf _{0} (x) completely known and for testing symmetry, i.e., testingH _{0}:f(x)=f(−x).
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 Title
 Nonparametric estimation of an affinity measure between two absolutely continuous distributions with hypotheses testing applications
 Journal

Annals of the Institute of Statistical Mathematics
Volume 32, Issue 1 , pp 223240
 Cover Date
 19801201
 DOI
 10.1007/BF02480327
 Print ISSN
 00203157
 Online ISSN
 15729052
 Publisher
 Kluwer Academic Publishers
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 Authors

 Ibrahim A. Ahmad ^{(1)}
 Author Affiliations

 1. McMaster University, Canada