Misspecification-Averse Estimation

TL;DR

This paper develops a framework for optimal estimation under likelihood misspecification, introducing a new criterion and characterizing asymptotically optimal estimators with theoretical guarantees.

Contribution

It introduces the constrained multiplier criterion for flexible misspecification attitudes and extends classical efficiency bounds to settings with moment-constrained misspecification.

Findings

Proves a local asymptotic minimax theorem for the new criterion.

Characterizes asymptotically optimal estimators as Bayes decision rules.

Shows feasible plug-in estimators are asymptotically optimal.

Abstract

We study optimal estimation when the likelihood may be misspecified. Building on tools from the theory of decision-making under uncertainty, we analyze a class of axiomatically grounded optimality criteria which nests several existing misspecification-robust objectives. Within this class, we introduce the constrained multiplier criterion, which allows for flexible misspecification attitudes. We prove a local asymptotic minimax theorem for this criterion, extending a classical efficiency bound to a limit experiment which incorporates moment-constrained misspecification concerns. We characterize asymptotically optimal estimators as Bayes decision rules under a flat prior and an exponentially tilted likelihood that incorporates the moment constraints, and show that feasible plug-in analogs are asymptotically optimal.