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Media type:
E-Article
Title:
A Simple Adaptive Estimator of the Integrated Square of a Density
Contributor:
Giné, Evarist;
Nickl, Richard
imprint:
International Statistics Institute / Bernoulli Society, 2008
Published in:Bernoulli
Language:
English
ISSN:
1350-7265
Origination:
Footnote:
Description:
<p>Given an i.i.d. sample <tex-math>$X_{1},\ldots ,X_{n}$</tex-math> with common bounded density f₀ belonging to a Sobolev space of order α over the real line, estimation of the quadratic functional <tex-math>$\int_{{\Bbb R}}f_{0}^{2}(x)\,{\rm d}x$</tex-math> is considered. It is shown that the simplest kernel-based plug-in estimator <tex-math>${\textstyle\frac{2}{n(n-1)h_{n}}}\sum_{1\leq i<j\leq n}k\left({\textstyle\frac{X_{i}-X_{j}}{h_{n}}}\right)$</tex-math> is asymptotically efficient if α > 1/4 and rate-optimal if α ≤ 1/4. A data-driven rule to choose the bandwidth <tex-math>$h_{n}$</tex-math> 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.</p>