Open Access
February 2018 Optimal adaptive inference in random design binary regression
Rajarshi Mukherjee, Subhabrata Sen
Bernoulli 24(1): 699-739 (February 2018). DOI: 10.3150/16-BEJ893


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.


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Rajarshi Mukherjee. Subhabrata Sen. "Optimal adaptive inference in random design binary regression." Bernoulli 24 (1) 699 - 739, February 2018.


Received: 1 January 2016; Revised: 1 July 2016; Published: February 2018
First available in Project Euclid: 27 July 2017

zbMATH: 06778345
MathSciNet: MR3706774
Digital Object Identifier: 10.3150/16-BEJ893

Keywords: adaptive confidence sets , binary regression , U-statistics

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 1 • February 2018
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