We study semiparametric models where for a fixed value of the finite-dimensional parameter there exists a sufficient statistic for the nuisance parameter. An asymptotically normal sequence of estimators for the parametric component is constructed, which is efficient under the assumption that projecting on the set of nuisance scores is equivalent to taking conditional expectations given the sufficient statistic. The latter property is checked for a number of examples, in particular for mixture models. We discuss the relation of our approach to conditional maximum likelihood estimation.
A. W. van der Vaart. "Estimating a Real Parameter in a Class of Semiparametric Models." Ann. Statist. 16 (4) 1450 - 1474, December, 1988. https://doi.org/10.1214/aos/1176351048