A simple adaptive estimator of the integrated square of a density

Details A simple adaptive estimator of the integrated square of a density A simple adaptive estimator of the integrated square of a density

2008-03-06

arxiv.org/abs/0803.0847v1

Bernoulli 2008, Vol. 14, No. 1, 47-61

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/07-BEJ110 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/07-BEJ110

Given an i.i.d. sample $X_1,...,X_n$ with common bounded density $f_0$ belonging to a Sobolev space of order $\alpha$ 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 i1/4$ and rate-optimal if $\alpha\le1/4$. A data-driven rule to choose the bandwidth $h_n$ is then proposed, which does not depend on prior knowledge of $\alpha$, so that the corresponding estimator is rate-adaptive for $\alpha \leq1/4$ and asymptotically efficient if $\alpha>1/4$.

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