Adaptive estimation in the single-index model via oracle approach

TL;DR

This paper introduces a new adaptive estimation method for single-index models that automatically adjusts to unknown parameters and function smoothness, achieving optimal rates in risk bounds.

Contribution

It proposes a novel procedure that adaptively estimates both the index vector and link function without prior knowledge, with proven optimal risk bounds.

Findings

Achieves local oracle inequality for pointwise risk

Establishes global oracle inequality under L_r norm

Demonstrates effectiveness for functions with varying smoothness

Abstract

In the framework of nonparametric multivariate function estimation we are interested in structural adaptation. We assume that the function to be estimated has the "single-index" structure where neither the link function nor the index vector is known. We suggest a novel procedure that adapts simultaneously to the unknown index and smoothness of the link function. For the proposed procedure, we prove a "local" oracle inequality (described by the pointwise seminorm), which is then used to obtain the upper bound on the maximal risk of the adaptive estimator under assumption that the link function belongs to a scale of H\"{o}lder classes. The lower bound on the minimax risk shows that in the case of estimating at a given point the constructed estimator is optimally rate adaptive over the considered range of classes. For the same procedure we also establish a "global" oracle inequality (under the $ L_r $ norm, $r< \infty $) and examine its performance over the Nikol'skii classes. This study shows that the proposed method can be applied to estimating functions of inhomogeneous smoothness, that is whose smoothness may vary from point to point.