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We consider the problem of uncertainty assessment for low dimensionalcomponents in high dimensional models. Specifically, we propose a decorrelatedscore function to handle the impact of high dimensional nuisance parameters. Weconsider both hypothesis tests and confidence regions for generic penalizedM-estimators. Unlike most existing inferential methods which are tailored forindividual models, our approach provides a general framework for highdimensional inference and is applicable to a wide range of applications. Fromthe testing perspective, we develop general theorems to characterize thelimiting distributions of the decorrelated score test statistic under both nullhypothesis and local alternatives. These results provide asymptotic guaranteeson the type I errors and local powers of the proposed test. Furthermore, weshow that the decorrelated score function can be used to construct point andconfidence region estimators that are semiparametrically efficient. We alsogeneralize this framework to broaden its applications. First, we extend it tohandle high dimensional null hypothesis, where the number of parameters ofinterest can increase exponentially fast with the sample size. Second, weestablish the theory for model misspecification. Third, we go beyond thelikelihood framework, by introducing the generalized score test based ongeneral loss functions. Thorough numerical studies are conducted to back up thedeveloped theoretical results.

A General Theory of Hypothesis Tests and Confidence Regions for Sparse High Dimensional Models