A simple adaptive estimator of the integrated square of a density

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 ∫_ℝf₀²(x) dx is considered. It is shown that the simplest kernel-based plug-in estimator 2n(n-1)hₙ ∑_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 $α>1/4$.