Universal near-optimality of CAVS over symmetric log-concave densities

Determine whether the constraint asymptotic variance selection (CAVS) estimator achieves near-optimal location-estimation rates, up to the stated polylogarithmic factors, universally for every symmetric log-concave density.

Background

The paper discusses the CAVS estimator, which selects a loss exponent in a data-dependent manner to interpolate among estimators such as the sample median, sample mean, and sample mid-range. Existing results establish near-optimality, modulo polylogarithmic factors, for a broad class of compactly supported symmetric location families satisfying certain moment constraints.

The unresolved issue is whether this near-optimality extends to the full class of symmetric log-concave densities, which includes both regular unbounded distributions and compactly supported distributions with substantially different instance-wise optimal rates. The question concerns universal performance across the entire shape-constrained class rather than performance under the more limited moment conditions previously analyzed.