Adaptive Wavelet Estimations in the Convolution Structure Density Model

Kaikai Cao,Xiaochen Zeng

doi:10.3390/math8091391

Abstract

Using kernel methods, Lepski and Willer study a convolution structure density model and establish adaptive and optimal Lp risk estimations over an anisotropic Nikol’skii space (Lepski, O.; Willer, T. Oracle inequalities and adaptive estimation in the convolution structure density model. Ann. Stat.2019, 47, 233–287). Motivated by their work, we consider the same problem over Besov balls by wavelets in this paper and first provide a linear wavelet estimate. Subsequently, a non-linear wavelet estimator is introduced for adaptivity, which attains nearly-optimal convergence rates in some cases.

Highlights

The estimation of a probability density from independent and identically distributed (i.i.d.)random observations X1, X2, · · ·, Xn of X is a classical problem in statistics
The L∞ risk optimal wavelet estimations were investigated by Lounici and Nickl [3]
When comparing the result of Theorem 2 with Theorem 1, we find that for the case r ≤ p, s0 p the convergence rate of non-linear estimator is better than that of the linear one with n s − 1r + 1p

Summary

Introduction

The estimation of a probability density from independent and identically distributed (i.i.d.). Provided L p (1 ≤ p ≤ ∞) risk optimal deconvolution estimations using wavelet bases. Based on the model (2) with some mild assumptions on Gα , Lepski and Willer [5] provided a lower bound estimation over L p risk on an anistropic Nikol’skii space. Risk estimations under the model (2) over Besov balls by using wavelets and expect to obtain the corresponding convergence rates. The convergence rates of Theorem 2 with the cases α = 0 and α = 1 are nearly-optimal (up to a logarithmic factor) by Donoho et al [1] and Li & Liu [4] respectively It is not clear whether the estimation in Theorem 2 is optimal (nearly-optimal) or not for α ∈ (0, 1).

Preliminaries

Proofs of Main Results

Conclusions