In this paper we consider the weighted average square error A,(rc)= (l/n)~=1 {f"(3))f(Xj)}2~(Xj), where f is the common density function of the independent and identically distributed random vectors X~ ..... X,, f, is the kernel estimator based on these vectors and ~z is a weight function. Using U-statistics techniques and the results of Gouri6roux and Tenreiro (Preprint 9617, Departamento de Matemfitica, Universidade de Coimbra, 1996), we establish a central limit theorem for the random variable A,(g) -EA,(Tz). This result enables us to compare the stochastic measures A,(~) and I,,0z. f) = f{f,(x)f(x)}2(g β’ f)(x)dx and to deduce an asymptotic expansion in probability for A,(lt) which extends a previous one, obtained, in a real context with it = 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 Koml6s-Major-Tusnfidy-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,(~z), and we study its local power properties.