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

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DOI: 10.1007/BF02480327

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

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## Abstract

Let*F* and*G* denote two distribution functions defined on the same probability space and are absolutely continuous with respect to the Lebesgue measure with probability density functions*f* and*g*, respectively. A measure of the closeness between*F* and*G* 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\(\hat \lambda = \left[ {\int {\hat f(x)dG_n (x) + \int {\hat g(x)dF_n (x)} } } \right]/\left[ {\int {\hat f^2 (x)dx + \int {\hat g^2 (x)dx} } } \right]\), where*F*_{n}*(x)* and*G*_{n}*(x)* are the empirical distribution functions of*F(x)* and*G(x)* respectively and\(\hat f(x)\) and\(\hat g(x)\) are taken to be the so-called kernel estimates of*f(x)* and*g(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., when*g(x)=f*_{0}*(x)*, with*f*_{0}*(x)* completely known and for testing symmetry, i.e., testing*H*_{0}:*f(x)=f(−x)*.