Kernel Density Estimators: Convergence in Distribution for Weighted Sup-norms

Let fn denote a kernel density estimator of a bounded continuous density f in the real line. Let Ψ(t) be a positive continuous function such that ‖Ψf‖∞<∞. Under natural smoothness conditions, necessary and sufficient conditions for the sequence √ nhn 2 log h−1 n supt∈R ∣∣Ψ(t)(fn(t)−Efn(t))∣∣ (properly centered and normalized) to converge in distribution to the double exponential law are obtained. The proof is based on Gaussian approximation and a (new) limit theorem for weighted sup-norms of a stationary Gaussian process. This extends well known results of Bickel and Rosenblatt to the case of weighted sup-norms, with the sup taken over the whole line. In addition, all other possible limit distributions of the above sequence are identified (subject to some regularity assumptions). This version: May, 2002 Runninghead: Kernel density estimators