Existence of Maximum Likelihood Estimates in Regression Models for Grouped and Ungrouped Data

Abstract

SUMMARY In general, concavity of the log likelihood alone does not imply that the MLE exists always. For a class of linear regression models for grouped and ungrouped data, a necessary and sufficient condition is obtained for the existence of the maximum likelihood estimator. This condition has an intuitively simple interpretation. Further, it turns out that there are similar necessary and sufficient conditions for the existence of maximum likelihood estimates for a number of other non-linear models such as Cox's Regression Model. For a given set of data, these conditions may be verified by Linear Programming Methods.