A Note on High-Dimensional Confidence Regions

TL;DR

This paper compares different high-dimensional confidence regions, showing that those based on all shapes can have exponentially smaller volume than hypercube-based regions, with validation through simulations.

Contribution

It introduces a comparison of all-based confidence regions with hypercube ones in high dimensions, highlighting volume efficiency.

Findings

all-based regions have exponentially smaller volume than hypercube regions.

Simulation results validate theoretical volume comparisons.

High-dimensional confidence regions can be optimized for volume efficiency.

Abstract

Recent advances in statistics introduced versions of the central limit theorem for high-dimensional vectors, allowing for the construction of confidence regions for high-dimensional parameters. In this note, $s$-sparsely convex high-dimensional confidence regions are compared with respect to their volume. Specific confidence regions which are based on $\ell_p$-balls are found to have exponentially smaller volume than the corresponding hypercube. The theoretical results are validated by a comprehensive simulation study.