Existence and Uniqueness of the Maximum Likelihood Estimator for a Multivariate Probit Model

Abstract

Abstract The multivariate probit model (MPM) is a particular case of the class of correlated prediction models. A correlated prediction model is especially useful when prediction or classification is envisaged into diagnostic classes that are combinations of binary responses. The parameter vector consists of a “location” part and an “association” part. The location part accounts for the effect the regressors have on the marginal probabilities of the binary responses. The association part corrects these probabilities, taking into account that the responses are related. This article investigates conditions for the existence and unicity of the maximum likelihood estimator (MLE) of the parameter vector. It turns out that the existence and uniqueness of the MLE for the location parameters when the association parameters are known are related to those of the multigroup logistic model. Necessary and sufficient conditions are given for the existence of the MLE of the association part. On the other hand the conditions for the unicity of the MLEs of the association parameters are much more complicated and not yet established. Finally, the article shows that for an MPM the estimates of the regression parameters for the location part exist and are unique if and only if they exist and are unique for each marginal univariate probit model. This result provides practical guidelines to detect early divergence Good starting values are essential; this problem is touched on briefly. The theoretical results are illustrated by a medical example.