Estimation adaptative dans le mod\`ele single-index par l'approche d'oracle

TL;DR

This paper introduces a new adaptive estimation method for single-index models that automatically adjusts to unknown parameters and smoothness, achieving optimal rates and applicability to functions with varying smoothness.

Contribution

It proposes a novel adaptive procedure for single-index models that simultaneously estimates the index and link function, with proven optimal risk bounds and applicability to inhomogeneous smoothness.

Findings

Establishes a local oracle inequality for the proposed estimator.

Provides an upper bound on the maximal risk under regularity assumptions.

Shows the estimator is minimax rate optimal over certain classes.

Abstract

In the framework of nonparametric multivariate function estimation we are interested in structural adaptation. We assume that the function to be estimated possesses the single-index structure where neither the link function nor the index vector is known. We propose a novel procedure that adapts simultaneously to the unknown index and smoothness of link function. For the proposed procedure, we present a "local" oracle inequality (described by the pointwise seminorm), which is then used to obtain the upper bound on the maximal risk under regularity assumption on the link function. The lower bound on the minimax risk shows that 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 study its performance over the Nikol'skii classes. This study shows that the proposed method can be applied to estimating functions of inhomogeneous smoothness.