Asymptotically Honest Confidence Regions for High Dimensional Parameters by the Desparsified Conservative Lasso

TL;DR

This paper introduces a desparsified conservative Lasso method for constructing asymptotically honest confidence regions in high-dimensional settings, improving accuracy and coverage over existing methods.

Contribution

It develops a novel desparsification approach for the conservative Lasso, allowing for honest confidence bands under weaker moment assumptions and heteroskedastic errors.

Findings

Desparsified conservative Lasso outperforms standard desparsified Lasso in parameter estimation.

The method provides better size control and coverage rates for confidence bands.

Simulation results confirm improved accuracy and reliability of the proposed approach.

Abstract

In this paper we consider the conservative Lasso which we argue penalizes more correctly than the Lasso and show how it may be desparsified in the sense of van de Geer et al. (2014) in order to construct asymptotically honest (uniform) confidence bands. In particular, we develop an oracle inequality for the conservative Lasso only assuming the existence of a certain number of moments. This is done by means of the Marcinkiewicz-Zygmund inequality. We allow for heteroskedastic non-subgaussian error terms and covariates. Next, we desparsify the conservative Lasso estimator and derive the asymptotic distribution of tests involving an increasing number of parameters. Our simulations reveal that the desparsified conservative Lasso estimates the parameters more precisely than the desparsified Lasso, has better size properties and produces confidence bands with superior coverage rates.