Minimax and minimax adaptive estimation in multiplicative regression : locally bayesian approach

TL;DR

This paper introduces a locally Bayesian minimax estimator for non-parametric multiplicative noise regression, achieving optimal adaptivity over isotropic Hölder classes through a novel combination of local polynomial fitting, Bayesian methods, and Lepski's technique.

Contribution

It develops a new adaptive estimation procedure that is minimax optimal in multiplicative regression models, with proven theoretical guarantees and simulation validation.

Findings

Estimator is minimax optimal over isotropic Hölder classes.

Proposed method adapts optimally using Lepski's technique.

Theoretical exponential inequality supports estimator's reliability.

Abstract

The paper deals with the non-parametric estimation in the regression with the multiplicative noise. Using the local polynomial fitting and the bayesian approach, we construct the minimax on isotropic H\"older class estimator. Next applying Lepski's method, we propose the estimator which is optimally adaptive over the collection of isotropic H\"older classes. To prove the optimality of the proposed procedure we establish, in particular, the exponential inequality for the deviation of locally bayesian estimator from the parameter to be estimated. These theoretical results are illustrated by simulation study.