Efficient maximum likelihood estimation in semiparametric mixture models

We consider maximum likelihood estimation in several examples of semiparametric mixture models, including the exponential frailty model and the errors-in-variables model. The observations consist of a sample of size n from the mixture density integral p(theta)(x\z)d eta(z). The mixing distribution is completely unknown. We show that the first component <(theta(n))over tilde> of the joint maximum likelihood estimator (<(theta(n))over tilde>, <(eta(n))over tilde>) is asymptotically normal and asymptotically efficient in the semiparametric sense.