Confidence Sets Based on Sparse Estimators Are Necessarily Large

TL;DR

This paper demonstrates that confidence sets derived from sparse estimators tend to be large, indicating a significant trade-off between sparsity and the precision of the confidence sets in parametric and semiparametric models.

Contribution

It establishes that sparsity in estimators inherently leads to larger confidence sets, highlighting a fundamental limitation of sparse estimation methods.

Findings

Confidence sets based on sparse estimators are necessarily large.

Sparsity incurs a substantial cost in the quality of confidence sets.

Results hold in general parametric and semiparametric frameworks.

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.