A simple adaptive estimator of the integrated square of a density

Abstract

Given an i.i.d. sample $X_1,...,X_n$ with common bounded density $f_0$ belonging to a Sobolev space of order $α$ over the real line, estimation of the quadratic functional $\int_{\mathbb{R}}f_0^2(x) \mathrm{d}x$ is considered. It is shown that the simplest kernel-based plug-in estimator \[\frac{2}{n(n-1)h_n}\sum_{1\leq i 1/4$ and rate-optimal if $α\le1/4$. A data-driven rule to choose the bandwidth $h_n$ is then proposed, which does not depend on prior knowledge of $α$, so that the corresponding estimator is rate-adaptive for $α\leq1/4$ and asymptotically efficient if $α>1/4$.