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June 2019 Bootstrap tuning in Gaussian ordered model selection
Vladimir Spokoiny, Niklas Willrich
Ann. Statist. 47(3): 1351-1380 (June 2019). DOI: 10.1214/18-AOS1717


The paper focuses on the problem of model selection in linear Gaussian regression with unknown possibly inhomogeneous noise. For a given family of linear estimators $\{\widetilde{\boldsymbol{{\theta}}}_{m},m\in\mathscr{M}\}$, ordered by their variance, we offer a new “smallest accepted” approach motivated by Lepski’s device and the multiple testing idea. The procedure selects the smallest model which satisfies the acceptance rule based on comparison with all larger models. The method is completely data-driven and does not use any prior information about the variance structure of the noise: its parameters are adjusted to the underlying possibly heterogeneous noise by the so-called “propagation condition” using a wild bootstrap method. The validity of the bootstrap calibration is proved for finite samples with an explicit error bound. We provide a comprehensive theoretical study of the method, describe in details the set of possible values of the selected model $\widehat{m}\in\mathscr{M}$ and establish some oracle error bounds for the corresponding estimator $\widehat{\boldsymbol{{\theta}}}=\widetilde{\boldsymbol{{\theta}}}_{\widehat{m}}$.


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Vladimir Spokoiny. Niklas Willrich. "Bootstrap tuning in Gaussian ordered model selection." Ann. Statist. 47 (3) 1351 - 1380, June 2019.


Received: 1 July 2015; Revised: 1 April 2018; Published: June 2019
First available in Project Euclid: 13 February 2019

zbMATH: 07053511
MathSciNet: MR3911115
Digital Object Identifier: 10.1214/18-AOS1717

Primary: 62G05
Secondary: 62G09 , 62J15

Keywords: oracle , Propagation condition , Smallest accepted

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.47 • No. 3 • June 2019
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