No. 1087

Title:

Impossibility of weak convergence of kernel density estimators to a tight non-degenerate law in L_2(R^d)

Author(s):

Nishiyama, Yoichi (The Institute of Statistical Mathematics)

Key words:

kernel estimator, weak convergence in Hilbert space

Abstract:

    Let $ \widehat{f}_n $ be the kernel estimator for the probability desity $ f $ on $ \BR^d $, and let $ f_n(x)=E\widehat{f}_n(x) $. It is shown that, for any sequence $ \{ r_n \} $ of positive constants such that $ r_n =o(\sqrt{n}) $, if $ r_n(\widehat{f}_{n}-f_n) $ converges weakly in $ L_2(\BR^d) $ to a tight Borel measurable random variable $ G $ then $ G $ is necessarily degenerate.