A simple adaptive estimator of the integrated square of a density
Giné, Evarist ; Nickl, Richard
Bernoulli, Tome 14 (2008) no. 1, p. 47-61 / Harvested from Project Euclid
Text on Project Euclid
PDF
Alt PDF
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 ¶ \[\frac{2}{n(n-1)h_{n}}\sum_{1\leq i\textless j\leq n}K\biggl(\frac {X_{i}-X_{j}}{h_{n}}\biggr)\] ¶ is asymptotically efficient if α>1/4 and rate-optimal if α≤1/4. A data-driven rule to choose the bandwidth 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.
Publié le : 2008-02-15
Classification:  adaptive estimation,  kernel density estimator,  quadratic functional 
@article{1202492784,
     author = {Gin\'e, Evarist and Nickl, Richard},
     title = {A simple adaptive estimator of the integrated square of a density},
     journal = {Bernoulli},
     volume = {14},
     number = {1},
     year = {2008},
     pages = { 47-61},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1202492784}
}

Giné, Evarist; Nickl, Richard. A simple adaptive estimator of the integrated square of a density. Bernoulli, Tome 14 (2008) no. 1, pp.  47-61. http://gdmltest.u-ga.fr/item/1202492784/