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

In this paper we consider the weighted average square error A(n)(pi)=(1/n)Sigma(j=1)(n ){f(n)(X-j)-f(X-j)}(2)pi(X-j), where f is the common density function of the independent and identically distributed random vectors X-1,...,X-n, f(n) is the kernel estimator based on these Vectors and x is a weight function. Using U-statistics techniques and the results of Gourieroux and Tenreiro (Preprint 9617, Departamento de Matematica, Universidade de Coimbra, 1996), we establish a central limit theorem for the random variable A(n)(pi) - EA(n)(pi). This result enables us to compare the stochastic measures A(n)(pi) and I-n(pi.f)= integral{f(n)(x)- f(x)}(2)(pi.f)(x) dx and to deduce an asymptotic expansion in probability for A(n)(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 Komlos-Major-Tusnady-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 A(n)(pi), and we study its local power properties. (C) 1998 Elsevier Science B.V. All rights reserved.