On the asymptotic behaviour of the integrated square error of kernel density estimators with data-dependent bandwidth

In this paper, we consider the integrated square error where f is the common density function of the independent and identically distributed random vectors X1,...,Xn and is the kernel estimator with a data-dependent bandwidth. Using the approach introduced by Hall (J. Multivariate Anal. 14 (1984) 1), and under some regularity conditions, we derive the L2 consistency in probability of and we establish an asymptotic expansion in probability and a central limit theorem for Jn.