  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.
