Optimal Adaptive Inference in Random Design Binary Regression

TL;DR

This paper develops adaptive confidence sets for nonparametric binary regression functions with unknown design density, identifying regimes where adaptation is possible and addressing related testing and estimation problems.

Contribution

It introduces a method for constructing adaptive confidence sets in binary regression with unknown design density, highlighting the conditions for successful adaptation.

Findings

Adaptive confidence sets are achievable under certain smoothness conditions.

Two key regimes determine the possibility of adaptation.

The approach addresses goodness of fit testing and parameter estimation.

Abstract

We construct confidence sets for the regression function in nonparametric binary regression with an unknown design density. These confidence sets are adaptive in $L^2$ loss over a continuous class of Sobolev type spaces. Adaptation holds in the smoothness of the regression function, over the maximal parameter spaces where adaptation is possible, provided the design density is smooth enough. We identify two key regimes --- one where adaptation is possible, and one where some critical regions must be removed. We address related questions about goodness of fit testing and adaptive estimation of relevant parameters.