Oleg Lepski

BANDWIDTH SELECTION IN KERNEL DENSITY ESTIMATION: ORACLE INEQUALITIES AND ADAPTIVE MINIMAX OPTIMALITY

We address the problem of density estimation with L s-loss by selection of kernel estimators. We develop a selection procedure and derive corresponding L s-risk oracle inequalities. It is shown that the proposed selection rule leads to the estimator being minimax adaptive over a scale of the anisotropic Nikolskii classes. The main technical tools used in our derivations are uniform bounds on the L s-norms of empirical processes developed recently by Goldenshluger and Lepski [Ann. Probab. (2011), to appear].

Thresholding algorithms, maxisets and well-concentrated bases

The aim of this paper is to synthetically analyse the performances of thresholding and wavelet estimation methods. In this connection, it is useful to describe the maximal sets where these methods attain a special rate of convergence. We relate these “maxisets” to other problems naturally arising in the context of non parametric estimation, as approximation theory or information reduction. A second part of the paper is devoted to isolate two very special properties especially shared by wavelet bases, which allow them to behave almost as in an Hilbertian context even for Lp risks.

Nonlinear estimation in anisotropic multi-index denoising

Abstract. In the framework of denoising a function depending of a multidimensional variable (for instance an image), we provide a nonparametric procedure which constructs a pointwise kernel estimation with a local selection of the multidimensional bandwidth parameter. Our method is a generalization of the Lepskis method of adaptation, and roughly consists in choosing the “coarsest” bandwidth such that the estimated bias is negligible. However, this notion becomes more delicate in a multidimensional setting. We will particularly focus on functions with inhomogeneous smoothness properties and especially providing a possible disparity of the inhomogeneous aspect in the different directions. We show, in particular that our method is able to exactly attain the minimax rate or to adapt to unknown degree of anisotropic smoothness up to a logarithmic factor, for a large scale of anisotropic Besov spaces.

Nonparametric estimation of composite functions

We study the problem of nonparametric estimation of a multivariate function g: R d → R that can be represented as a composition of two unknown smooth functions f: R → R and G: R d → R. We suppose that f and G belong to known smoothness classes of functions, with smoothness γ and β, respectively. We obtain the full description of minimax rates of estimation of g in terms of γ and β, and propose rate-optimal estimators for the sup-norm loss. For the construction of such estimators, we first prove an approximation result for composite functions that may have an independent interest, and then a result on adaptation to the local structure. Interestingly, the construction of rate-optimal estimators for composite functions (with given, fixed smoothness) needs adaptation, but not in the traditional sense: it is now adaptation to the local structure. We prove that composition models generate only two types of local structures: the local single-index model and the local model with roughness isolated to a single dimension (i.e., a model containing elements of both additive and single-index structure). We also find the zones of (γ, β) where no local structure is generated, as well as the zones where the composition modeling leads to faster rates, as compared to the classical nonparametric rates that depend only to the overall smoothness of g.

Multivariate density estimation under sup-norm loss: Oracle approach, adaptation and independence structure

This paper deals with the density estimation on R d under sup-norm loss. We provide a fully data-driven estimation procedure and establish for it a so-called sup-norm oracle inequality. The proposed estimator allows us to take into account not only approximation properties of the underlying density, but eventual independence structure as well. Our results contain, as a particular case, the complete solution of the bandwidth selection problem in the multi-variate density model. Usefulness of the developed approach is illustrated by application to adaptive estimation over anisotropic Nikolskii classes.

Adaptive estimation over anisotropic functional classes via oracle approach

We address the problem of adaptive minimax estimation in white gaussian noise model under

Lower bounds in the convolution structure density model

L_p

Upper functions for

--loss,

\mathbb{L}_{p}

1\leq p\leq\infty,

-norms of Gaussian random fields

on the anisotropic Nikolskii classes. We present the estimation procedure based on a new data-driven selection scheme from the family of kernel estimators with varying bandwidths. For proposed estimator we establish so-called Lp-norm oracle inequality and use it for deriving minimax adaptive results. We prove the existence of rate-adaptive estimators and fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm indexes in definitions of the functional class and of the risk. In particular some new asymptotics of the minimax risk are discovered including necessary and sufficient conditions for existence a uniformly consistent estimator. We provide also with detailed overview of existing methods and results and formulate open problems in adaptive minimax estimation.