Stochastic Ordering under Weaker Likelihood-Ratio Shape Conditions

TL;DR

This paper demonstrates that likelihood ratio shape assumptions can be relaxed while still maintaining endpoint criteria for stochastic orders, broadening applicability especially for discontinuous likelihood ratios.

Contribution

It introduces weaker conditions on likelihood ratios, such as unimodality and sign-pattern constraints, to preserve stochastic order criteria.

Findings

Shape hypotheses can be weakened while retaining endpoint criteria.

Unimodality and sign-pattern conditions suffice for stochastic order preservation.

Applicable to discontinuous likelihood ratios, expanding practical use cases.

Abstract

We show that the shape hypothesis on a likelihood ratio can be weakened while retaining endpoint criteria for the hazard-rate and usual stochastic orders. The endpoint reduction persists under unimodality of the likelihood ratio and under a sign-pattern condition on the likelihood ratio minus one, with at most two sign changes and a negative right tail. It also follows from a direct superlevel-set criterion involving the same expression, which is useful in particular for discontinuous likelihood ratios.