A Note on High-Dimensional Confidence Regions

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 ${\mathrm{\ell }}_p$-balls are found to have exponentially smaller volume than the corresponding hypercube. The theoretical results are validated by a comprehensive simulation study.