On the Symmetry of Limiting Distributions of M-estimators

TL;DR

This paper investigates the symmetry properties of limiting distributions of M-estimators, facilitating inference methods like HulC even when the distributions are non-normal, thus broadening the scope of statistical inference.

Contribution

It establishes conditions under which the limiting distributions of M-estimators are symmetric, extending inference techniques beyond the normal approximation.

Findings

Identifies conditions for symmetry of limiting distributions

Enables inference with non-normal limits using HulC

Broadens applicability of asymptotic inference methods

Abstract

Many functionals of interest in statistics and machine learning can be written as minimizers of expected loss functions. Such functionals are called $M$-estimands, and can be estimated by $M$-estimators -- minimizers of empirical average losses. Traditionally, statistical inference (e.g., hypothesis tests and confidence sets) for $M$-estimands is obtained by proving asymptotic normality of $M$-estimators centered at the target. However, asymptotic normality is only one of several possible limiting distributions and (asymptotically) valid inference becomes significantly difficult with non-normal limits. In this paper, we provide conditions for the symmetry of three general classes of limiting distributions, enabling inference using HulC (Kuchibhotla et al. (2024)).