Efficiency Bound Under Identifiability Constraints in Semiparametric Models

The purpose of this work is to define an adequate efficiency bound in some models presenting some identification problems. We show how it is possible to define such bounds in some regular semi-parametric models (in the sense of Le Cam) when an identifying constraint is available, despite the degeneracy of the information matrix. We establish a new convolution theorem in this context. We illustrate the computation of the information bound for some standard identifiability constraints, in some interesting models, including probit, single-index, and ANOVA models. We also show how a two-step procedure still based on a preliminary estimator satisfying approximately the constraint, allows us to obtain an efficient estimator of the parameters.