Suppose $F$ is a mixture of a known parametric family of distributions with an unknown nonparametric mixing distribution. Consider the problem of estimating the value $T(dF)$ of a smooth functional $T$ at the mixed distribution $F$. One nonparametric estimator of $T(dF)$ is $T(dF_n)$ where $F_n$ denotes the empirical distribution. Under a differentiability condition on $T$ given in this note, $T(dF_n)$ is shown to be fully efficient asymptotically in the sense that the limiting distribution of any other estimator of $T(dF)$ that is regular in the sense of Hajek and Beran can be expressed as a convolution of the limiting normal distribution of $T(dF_n)$ with another distribution. The differentiability condition is verified for the case that the parametric family being mixed is a one-parameter exponential family and the support of the mixing distribution is an infinite set with a finite accumulation point. Regularity is verified for the maximum likelihood estimator of the probability of zero in a mixed Poisson distribution under certain conditions on the mixing distribution. Moreover, the maximum likelihood estimator and the sample proportion of zeroes are both shown to be fully efficient in this example.
"Asymptotic Efficiency of Estimators of Functionals of Mixed Distributions." Ann. Statist. 12 (4) 1380 - 1387, December, 1984. https://doi.org/10.1214/aos/1176346798