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.

翻译：我们刻画了作为一类有限混合模型推断函数的得分函数的无偏性。所研究的模型代表了观测值被分层为有限个组的情形。我们证明，在温和正则条件下，用于估计每个组分布参数的得分函数是无偏的。我们还证明，如果在上述场景中引入混合，使得某些观测值仅以不属于{0,1}的概率属于某些组，则得分函数会变得有偏。我们进而论证，在进一步温和正则条件下，极大似然估计是不一致的。上述结果可推广至包含任意冗余参数的正则模型，包括半参数模型。