Extend the overlap criterion to nonlinear threshold maps

Extend the overlap criterion for finite maximum-likelihood-estimator existence in latent polyhedral categorical response models to categorical response models whose comparison thresholds are nonlinear functions of the parameter, including reference-ratio models.

Background

The paper establishes that, for latent polyhedral categorical response models with thresholds linear in the parameter, overlap of the pooled structure-vector set characterizes the existence of a finite maximum likelihood estimate under suitable continuity and strict threshold-monotonicity conditions. Reference-ratio models are identified as lying outside this framework because their comparison thresholds depend nonlinearly on the parameter. The authors explicitly identify extending the criterion beyond linear threshold maps as the natural next research problem.