Universal pointwise selection rule in multivariate function estimation

In this paper, we study the problem of pointwise estimation of a multivariate function. We develop a general pointwise estimation procedure that is based on selection of estimators from a large parameterized collection. An upper bound on the pointwise risk is established and it is shown that the proposed selection procedure specialized for different collections of estimators leads to minimax and adaptive minimax estimators in various settings.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ144 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Estimation in the convolution structure density model. Part I: oracle inequalities

We study the problem of nonparametric estimation under \bL_p-loss, $p\in [1,\infty)$, in the framework of the convolution structure density model on \bR^d. This observation scheme is a generalization of two classical statistical models, namely density estimation under direct and indirect observations. In Part I the original pointwise selection rule from a family of "kernel-type" estimators is proposed. For the selected estimator, we prove an \bL_p-norm oracle inequality and several of its consequences. In Part II the problem of adaptive minimax estimation under \bL_p--loss over the scale of anisotropic Nikol'skii classes is addressed. We fully characterize the behavior of the minimax risk for different relationships between regularity parameters and norm indexes in the definitions of the functional class and of the risk. We prove that the selection rule proposed in Part I leads to the construction of an optimally or nearly optimally (up to logarithmic factor) adaptive estimator

Estimation in the convolution structure density model. Part II: adaptation over the scale of anisotropic classes

This paper continues the research started in \cite{LW16}. In the framework of the convolution structure density model on \bR^d, we address the problem of adaptive minimax estimation with \bL_p--loss over the scale of anisotropic Nikol'skii classes. We fully characterize the behavior of the minimax risk for different relationships between regularity parameters and norm indexes in the definitions of the functional class and of the risk. In particular, we show that the boundedness of the function to be estimated leads to an essential improvement of the asymptotic of the minimax risk. We prove that the selection rule proposed in Part I leads to the construction of an optimally or nearly optimally (up to logarithmic factor) adaptive estimator

On adaptive minimax density estimation on RdR^d

We address the problem of adaptive minimax density estimation on \bR^d with \bL_p--loss on the anisotropic Nikol'skii classes. We 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, we show that there are four different regimes with respect to the behavior of the minimax risk. We develop a single estimator which is (nearly) optimal in orderover the complete scale of the anisotropic Nikol'skii classes. Our estimation procedure is based on a data-driven selection of an estimator from a fixed family of kernel estimators

Upper functions for positive random functionals. Application to the empirical processes theory II

International audienceThis part of the paper finalizes the research started in Lepski (2013b)

ADAPTIVE ESTIMATION OVER ANISOTROPIC FUNCTIONAL CLASSES VIA ORACLE APPROACH

International audienceWe address the problem of adaptive minimax estimation in white Gaus-sian noise models under L p-loss, 1 ≤ p ≤ ∞, on the anisotropic Nikol'skii classes. We present the estimation procedure based on a new data-driven selection scheme from the family of kernel estimators with varying bandwidths. For the proposed estimator we establish so-called L p-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 the existence of a uniformly consistent estimator. We provide also a detailed overview of existing methods and results and formulate open problems in adaptive minimax estimation