Optimal adaptive inference in random design binary regression

Abstract

We construct confidence sets for the regression function in nonparametric binary regression with an unknown design density – a nuisance parameter in the problem. 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 infinite dimensional parameters.