Almost Sure Uniqueness of a Global Minimum Without Convexity

2018/03/06 by Cox, Gregory
#Econometrics (econ.EM) #FOS: Economics and business #FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1803.02415

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.

Related

• An invitation to the study of a uniqueness problem
• On semi-continuity and continuity of the smallest and largest minimizing point of real convex functions with applications in probability and statistics
• Most Convex Functions Have Unique Minimizers
• Convex Optimization on Functionals of Probability Densities
• Regularly random duality