Adaptive estimation under single-index constraint in a regression model

TL;DR

This paper introduces an adaptive estimation method for multivariate functions in single-index models, effectively estimating both the unknown index and the link function's smoothness under weak noise assumptions.

Contribution

It proposes a novel kernel-based estimator that adaptively estimates the index vector and the link function's smoothness in a single-index regression model.

Findings

Establishes a pointwise oracle inequality for the estimator.

Derives a global oracle inequality for entire function estimation.

Demonstrates adaptivity over Hölder and Nikol'skii classes.

Abstract

The problem of adaptive multivariate function estimation in the single-index regression model with random design and weak assumptions on the noise is investigated. A novel estimation procedure that adapts simultaneously to the unknown index vector and the smoothness of the link function by selecting from a family of specific kernel estimators is proposed. We establish a pointwise oracle inequality which, in its turn, is used to judge the quality of estimating the entire function (``global'' oracle inequality). Both the results are applied to the problems of pointwise and global adaptive estimation over a collection of H\"{o}lder and Nikol'skii functional classes, respectively.