Functional asymptotic normality of the L2-deviation of the kernel density estimation indexed by classes of weight functions

The goal of this work is to establish the functional asymptotic normality of the L2-deviation of the kernel density estimator fn indexed by a family W of weight functions W defined byIn(W):=∫A(fn(x)-Efn(x))2W(x)dx,W∈W,where f is the common density function of the i.i.d. real-valued sample X1,…,Xn on which fn is calculated and A is a Borel subset of R. We apply this result to derive new statistics in the spirit of Bickel-Rosenblatt (BR) to test goodness-of-fit of the density function f. These new tests are more powerful than BR test under some local alternatives.