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 $∫_ℝf_0^2(x) \mathrm{d}x$ is considered. It is shown that the simplest kernel-based plug-in estimator $$\frac{2}{n(n-1)h_{n}} \sum_{1\leq i<j\leq n} K\biggl(\frac {X_{i}-X_{j}}{h_{n}}\biggr)$$ is asymptotically efficient if $α>1/4$ and rate-optimal if $α≤1/4$. 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 $α≤1/4$ and asymptotically efficient if $α>1/4$.