BOOTSTRAP TUNING IN GAUSSIAN ORDERED MODEL SELECTION

• 成果类型: Article
• 署名作者: Spokoiny, Vladimir; Willrich, Niklas
• 署名单位: Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; Humboldt University of Berlin; Russian Academy of Sciences; HSE University (National Research University Higher School of Economics)
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1717
• 发表日期: 2019
• 页码: 1351-1380
• 关键词: minimax adaptive estimation; resampling methods; adaptation; jackknife; bounds

摘要

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 {(theta) over tilde (m), m is an element of 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 (m) over cap is an element of M and establish some oracle error bounds for the corresponding estimator (theta) over cap = (theta) over tilde ((m) over cap).