Rare Event Analysis for Minimum Hellinger Distance Estimators via Large Deviation Theory

Hellinger distance has been widely used to derive objective functions that are alternatives to maximum likelihood methods. While the asymptotic distributions of these estimators have been well investigated, the probabilities of rare events induced by them are largely unknown. In this article, we analyze these rare event probabilities using large deviation theory under a potential model misspecification, in both one and higher dimensions. We show that these probabilities decay exponentially, characterizing their decay via a rate function which is expressed as a convex conjugate of a limiting cumulant generating function. In the analysis of the lower bound, in particular, certain geometric considerations arise that facilitate an explicit representation, also in the case when the limiting generating function is nondifferentiable. Our analysis involves the modulus of continuity properties of the affinity, which may be of independent interest.

Automated summary

This article analyzes the probabilities of rare events induced by Hellinger distance using large deviation theory under potential model misspecifications. The probabilities are found to decay exponentially, with the decay characterized by a rate function expressed as a convex conjugate of a limiting cumulant generating function. Geometric considerations arising from the analysis facilitate an explicit representation, even in cases where the limiting generating function is nondifferentiable.