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

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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 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)=f 0 (x), withf 0 (x) completely known and for testing symmetry, i.e., testingH 0:f(x)=f(−x).