Existence of the MLE and propriety of posteriors for a general multinomial choice model

This paper examines necessary and sufficient conditions for the existence of Maximum Likelihood Estimates (MLE) and the propriety of the posterior under a bounded improper prior density for a wide class of discrete (or multinomial) choice models. The choice models are based on the principle of utility maximization. Our results cover a wide class of latent variable distributions defining the utility, including in particular multinomial logistic and probit classification and choice models as special cases. Albert and Anderson (1984) gave separation and overlap conditions for the existence of the MLE in logistic classification models. We generalize their conditions to multinomial choice models, giving necessary and sufficient conditions for the existence of a finite MLE and the propriety of the posterior for a wide class of bounded improper priors. Consistency and asymptotic normality for both the MLE and the posterior are also proved under mild conditions.