Universal pointwise selection rule in multivariate function estimation

TL;DR

This paper introduces a universal pointwise estimation method for multivariate functions, utilizing a selection procedure from a large estimator collection, achieving minimax and adaptive minimax optimality.

Contribution

It develops a general selection-based estimation procedure that adapts to different settings, providing minimax optimality in multivariate function estimation.

Findings

Establishes an upper bound on pointwise risk.

Demonstrates minimax and adaptive minimax properties.

Applicable to various estimator collections.

Abstract

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.