On the Bias of the Score Function of Finite Mixture Models

Abstract

We characterize the unbiasedness of the score function, viewed as an inference function, for a class of finite mixture models. The models studied represent the situation where there is a stratification of the observations in a finite number of groups. We show that if the observations belonging to the same group follow the same distribution and the K distributions associated with each group are distinct elements of a sufficiently regular parametric family of probability measures, then the score function for estimating the parameters identifying the distribution of each group is unbiased. However, if one introduces a mixture in the scenario described above, so that for some observations it is only known that they belong to some of the groups with a given probability (not all in { 0, 1}), then the score function becomes biased. We argue then that under further mild regularity conditions, the maximum likelihood estimate is not consistent.Comment: 9 page