Median Regularity and Honest Inference

TL;DR

This paper introduces median regularity, a new estimator property, and proves it is both necessary and sufficient for honest, uniformly valid inference of a functional, filling a gap in the existing literature.

Contribution

The paper defines median regularity and establishes its equivalence to the possibility of honest inference, a novel concept not previously documented.

Findings

Median regularity is necessary and sufficient for honest inference.

A new theoretical framework linking estimator properties to inference validity.

Fills a gap in the literature on estimator regularity and inference.

Abstract

We introduce a new notion of regularity of an estimator called median regularity. We prove that uniformly valid (honest) inference for a functional is possible if and only if there exists a median regular estimator of that functional. To our knowledge, such a notion of regularity that is necessary for uniformly valid inference is unavailable in the literature.