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

Abstract

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)=f₀(x), withf₀(x) completely known and for testing symmetry, i.e., testingH₀:f(x)=f(−x).