Oracle inequalities and adaptive estimation in the convolution structure density model

Abstract

We study the problem of nonparametric estimation under Lp-loss, p ∈ [1, ∞), in the framework of the convolution structure density model on R d . This observation scheme is a generalization of two classical statistical models, namely density estimation under direct and indirect observations. The original pointwise selection rule from a family of "kernel-type" estimators is proposed. For the selected estimator, we prove an Lp-norm oracle inequality and several of its consequences. Next, the problem of adaptive minimax estimation under Lp-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 proposed selection rule leads to the construction of an optimally or nearly optimally (up to logarithmic factor) adaptive estimator.

Density deconvolution under general assumptions on the distribution of measurement errors

“…The problem of density deconvolution has been extensively studied in the literature; see, e.g., Carroll & Hall (1988), Zhang (1990), Fan (1991), Butucea & Tsybakov (2008a, 2008b, Lounici & Nickl (2011), Comte & Lacour (2013) and Lepski & Willer (2019). We also refer to the book of Meister (2009), where many additional references can be found.…”

Minimax bounds for Besov classes in density estimation

“…We suspect, however, that already existing estimators are (near) minimax. Of particular note are the wavelet estimators of [BTWB10, RBRTM11] and the selection rule of [LW19]. Unfortunately, the authors did not explicitly address this issue.…”

“…What is missing is the computation of the risk from these inequalities. As for the very general paper of [LW19], the authors explain in their Theorem 2 how to control the maximal risk of their estimator over a collection F. The functions of F need not to be bounded. The condition relates rather to the inclusion of F into a L q space or more generally a Lorentz space.…”

“…For each of these references, a logarithmic factor in the convergence rates is to be expected. In [LW19], this factor may be due to a too local approach to the estimation problem. In our paper, and in [BTWB10,RBRTM11], this factor appears in the penalties/thresholds.…”

“…They lead to nice oracle inequalities for the L 2 loss without condition on the supremum norm of f . Also worth mentioning are the general procedures of [LW19,Bir06,Bir14]. The first is based on a pointwise selection scheme.…”