Asymptotic distribution for a discrete version of integrated square error of multivariate density kernel estimators

Abstract

In this paper we consider the weighted average square error An([pi]) = (1/n) [Sigma]nj=1{fn(Xj) -; f(Xj)}2[pi](Xj), where f is the common density function of the independent and identically distributed random vectors X1,..., Xn, fn is the kernel estimator based on these vectors and [pi] is a weight function. Using U-statistics techniques and the results of Gouriéroux and Tenreiro (Preprint 9617, Departamento de Matemática, Universidade de Coimbra, 1996), we establish a central limit theorem for the random variable An([pi]) -; EAn([pi]). This result enables us to compare the stochastic measures An([pi]) and In([pi] · f) = [integral operator]{fn(x) -; f(x)}2([pi] · f)(x)dx and to deduce an asymptotic expansion in probability for An([pi]) which extends a previous one, obtained, in a real context with [pi] = 1, by Hall (Stochastic Processes and their Applications, 14 (1982) pp. 1-16). The approach developed in this paper is different from the one adopted by Hall, since he uses Komls-Major-Tusnády-type approximations to the empiric distribution function. Finally, applications to goodness-of-fit tests are considered. More precisely, we present a consistent test of goodness-of-fit for the functional form of f based on a corrected bias version of An([pi]), and we study its local power properties. © 1998 Elsevier Science B.V. All rights reserved.