A simple adaptive estimator of the integrated square of a density

Abstract

Given an i.i.d. sample X1,..., Xn with common bounded density f0 belonging to a Sobolev space of order α over the real line, estimation of the quadratic functional fℝ f02 (x) dx is considered. It is shown that the simplest kernel-based plug-in estimator 2/n(n - 1)hn σm 1≤i 1/4 and rate-optimal if α < 1/4. A data-driven rule to choose the bandwidth hn is then proposed, which does not depend on prior knowledge of α, so that the corresponding estimator is rate-adaptive for α ≤ 1/4 and asymptotically efficient if α > 1/4. © 2008 ISI/BS.