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A general theory of hypothesis tests and confidence regions for sparse high dimensional models

Author(s): Ning, Yang; Liu, Han

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Abstract: We consider the problem of uncertainty assessment for low dimensional components in high dimensional models. Specifically, we propose a decorrelated score function to handle the impact of high dimensional nuisance parameters. We consider both hypothesis tests and confidence regions for generic penalized M-estimators. Unlike most existing inferential methods which are tailored for individual models, our approach provides a general framework for high dimensional inference and is applicable to a wide range of applications. From the testing perspective, we develop general theorems to characterize the limiting distributions of the decorrelated score test statistic under both null hypothesis and local alternatives. These results provide asymptotic guarantees on the type I errors and local powers of the proposed test. Furthermore, we show that the decorrelated score function can be used to construct point estimators that are semi-parametrically efficient and the optimal confidence regions. We also generalize this framework to broaden its applications. First, we extend it to handle high dimensional null hypothesis,where the number of parameters of interest can increase exponentially fast with the sample size. Second, we establish the theory for model misspecification. Third, we go beyond the likelihood framework, by introducing the generalized score test based on general loss functions. Thorough numerical studies are conducted to back up the developed theoretical results.
Publication Date: Feb-2017
Citation: Ning, Yang, Liu, Han. (2017). A general theory of hypothesis tests and confidence regions for sparse high dimensional models. The Annals of Statistics, 45 (1), 158 - 195. doi:10.1214/16-AOS1448
DOI: doi:10.1214/16-AOS1448
ISSN: 0090-5364
Pages: 158 - 195
Type of Material: Journal Article
Journal/Proceeding Title: The Annals of Statistics
Version: Author's manuscript

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