Confidence Sets Based on Sparse Estimators Are Necessarily Large

Abstract

Confidence sets based on sparse estimators are shown to be large compared to more standard confidence sets, demonstrating that sparsity of an estimator comes at a substantial price in terms of the quality of the estimator. The results are set in a general parametric or semiparametric framework.

Revision containing correction of some minor errors and typos; some additional remarks added

References in corpus (3)

• Composite quantile regression and the oracle Model Selection Theory
• Variable selection in semiparametric regression modeling
• Sparse Estimators and the Oracle Property, or the Return of Hodges' Estimator

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• Confidence Sets Based on Penalized Maximum Likelihood Estimators in Gaussian Regression
• On the Post Selection Inference constant under Restricted Isometry Properties
• On Hodges' Superefficiency and Merits of Oracle Property in Model Selection
• On the Distribution of the Adaptive LASSO Estimator
• Admissibility of the usual confidence set for the mean of a univariate or bivariate normal population: The unknown-variance case
• Uniformly Valid Inference Based on the Lasso in Linear Mixed Models
• Uniform Asymptotics and Confidence Regions Based on the Adaptive Lasso with Partially Consistent Tuning