Almost Sure Uniqueness of a Global Minimum Without Convexity

TL;DR

This paper proves that the global minimum of a random objective function is almost surely unique without requiring convexity, with applications in statistics such as M-estimators and threshold regression.

Contribution

It introduces a general theoretical result establishing almost sure uniqueness of the argmin without convexity, applicable to various statistical estimation problems.

Findings

Almost sure uniqueness of the global minimum without convexity.

Application to M-estimators and penalized likelihood estimators.

Results on threshold regression and weak identification.

Abstract

This paper establishes the argmin of a random objective function to be unique almost surely. This paper first formulates a general result that proves almost sure uniqueness without convexity of the objective function. The general result is then applied to a variety of applications in statistics. Four applications are discussed, including uniqueness of M-estimators, both classical likelihood and penalized likelihood estimators, and two applications of the argmin theorem, threshold regression and weak identification.