We address the problem of density estimation with Ls-loss by selection of kernel estimators. We develop a selection procedure and derive corresponding Ls-risk oracle inequalities. It is shown that the proposed selection rule leads to the estimator being minimax adaptive over a scale of the anisotropic Nikol'skii classes. The main technical tools used in our derivations are uniform bounds on the Ls-norms of empirical processes developed recently by Goldenshluger and Lepski

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Let a Hölder continuous function f be observed with noise.In the present paper we study the problem of nonparametric estimation of certain nonsmooth functionals of f, specifically, L r norms f r of f . Known from the literature results on functional estimation deal mostly with two extreme cases: estimating a smooth (differentiable in L 2 ) functional or estimating a singular functional like the value of f at certain point or the maximum of f. In the first case, the convergence rate typically is n −1/2 , n being the number of observations. In the second case, the rate of convergence coincides with the one of estimating the function f itself in the corresponding norm.We show that the case of estimating f r is in some sense intermediate between the above extremes. The optimal rate of convergence is worse than n −1/2 but is better than the rate of convergence of nonparametric estimates of f . The results depend on the value of r. For r even integer, the rate occurs to be n −β/(2β+1−1/r) where β is the degree of smoothness. If r is not an even integer, then the nonparametric rate n −β/(2β+1) can be improved, but only by a logarithmic in n factor. Mathematics Subject Classification (1991): 62G07; Secondary 62G20Key words: Non-smooth functionals -Integral norm -Rate of estimationThe thoughtful comments of two anonymous referees, leading to significant improvement of this paper, are gratefully acknowledged. The authors thank A. Tsybakov and Yu. Golubev for important remarks and discussion.

Approximation methods, Besov spaces, denoising, minimax rate of convergence, non parametric estimation, oracle inequalities, saturation spaces, wavelet thresholding, 62G05, 62G07, 62G20,

The problem of optimal adaptive estimation of a function at a given point from noisy data is considered. Two procedures are proved to be asymptotically optimal for different settings.First we study the problem of bandwidth selection for nonparametric pointwise kernel estimation with a given kernel. We propose a bandwidth selection procedure and prove its optimality in the asymptotic sense. Moreover, this optimality is stated not only among kernel estimators with a variable bandwidth. The resulting estimator is asymptotically optimal among all feasible estimators. The important feature of this procedure is that it is fully adaptive and it "works" for a very wide class of functions obeying a mild regularity restriction. With it the attainable accuracy of estimation depends on the function itself and is expressed in terms of the "ideal adaptive bandwidth" corresponding to this function and a given kernel.The second procedure can be considered as a specialization of the first one under the qualitative assumption that the function to be estimated belongs to some Hölder class β L with unknown parameters β L. This assumption allows us to choose a family of kernels in an optimal way and the resulting procedure appears to be asymptotically optimal in the adaptive sense in any range of adaptation with β ≤ 2.