Two-step combined nonparametric likelihood estimation of misspecified semiparametric models

This paper proposes to estimate possibly misspecified semiparametric estimating equations models using a two-step combined nonparametric likelihood method. The method uses in the first step the plug in principle and replaces the infinite dimensional parameter with a consistent estimator. In the second step an estimator for the finite dimensional parameter is obtained by combining exponential tilting with a another member of the empirical Cressie-Read discrepancy. The resulting class of semiparametric estimators are robust to misspecification and have the same asymptotic variance as that of the efficient semiparametric generalised method of moment estimator under correct specification. It is also shown that the asymptotic distributions of the proposed estimators can be consistently estimated by a multiplier bootstrap procedure. The results of the paper are illustrated with a quadratic inference function model and an instrumental variable partially linear additive model. Monte Carlo evidence suggests that the proposed estimators have competitive finite sample properties.