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Exact adaptive pointwise estimation on Sobolev classes of densities

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ps72.dvi ESAIM: Probability and Statistics May 2001, Vol. 5, 1{31 URL: EXACT ADAPTIVE POINTWISE ESTIMATION ON SOBOLEV CLASSES OF DENSITIES Cristina Butucea1 Abstract. The subject of this paper is to estimate adaptively the common probability density of n independent, identically distributed random variables. The estimation is done at a �xed point x0 2 R, over the density functions that belong to the Sobolev class Wn(�;L). We consider the adaptive problem setup, where the regularity parameter � is unknown and varies in a given set Bn. A sharp adaptive estimator is obtained, and the explicit asymptotical constant, associated to its rate of convergence is found. Mathematics Subject Classi�cation. 62N01, 62N02, 62G20. Received November 15, 2000. Revised April 9, 2001. 1. Introduction Consider n independent, identically distributed random variables X1; : : : ; Xn, having common unknown probability density f : R! [0;1). We assume that f belongs to a Sobolev class of densities. For any L > 0 and � positive integer, we de�ne the Sobolev class of densities W (�; L), as the set of functions W (�; L) = � f : R! [0;1) : Z R f = 1; Z R � f (�) (x) �2 dx � L2 � ; where f (�) will denote from now on the generalized derivative of order � of f . We may de�ne for an absolutely integrable function f : R! R its Fourier transform F(f)(x) = RR f(y)e−ixy dy, for any x in R. We adopt now a more general de�nition of the Sobolev class, allowing non-integer values of � > 1=2 W (�; L) = � f : R! [0;1) : Z R f = 1; Z R jF (f) (x)j2 jxj2� dx � 2�L2 � � Let fn be an estimator of f based on the sample X1; : : : ; Xn and x0 a �xed point. The performance of the estimator fn at the point x0 is measured by the maximal risk R�n;�(fn; ’n;�) = sup f2W (�;L) Ef h ’−qn;� jfn(x0)− f(x0)j q i ; (1.1) Keywords and phrases: Density estimation, exact asymptotics, pointwise risk, sharp adaptive estimator. 1 Universit�e Paris 10, Modal’X, ba^timent G, 200 avenue de la R�epub

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