Suppose one observes independent samples of size $n$ from both the mixture density $\int p(x \mid z) d\eta(z)$ and from the distribution $\eta$. The kernel $p(x \mid z)$ is known. We show asymptotic normality and efficiency of the maximum likelihood estimator for $\eta$.
"Maximum Likelihood Estimation with Partially Censored Data." Ann. Statist. 22 (4) 1896 - 1916, December, 1994. https://doi.org/10.1214/aos/1176325763