On the Bias of the Score Function of Finite Mixture Models

We characterise 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, under mild regularity conditions, the score function for estimating the parameters identifying each group's distribution is unbiased. We also show that 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 probability not in $\{ 0, 1 \}$, then the score function becomes biased. We argue then that under further mild regularity, the maximum likelihood estimate is not consistent. The results above are extended to regular models containing arbitrary nuisance parameters, including semiparametric models.