A PUZZLING PHENOMENON IN SEMIPARAMETRIC ESTIMATION PROBLEMS WITH INFINITE-DIMENSIONAL NUISANCE PARAMETERS

This note considers a puzzling phenomenon that is observed in some semiparametric estimation problems+ In some cases, using estimated values of the nuisance parameters provides a more efficient estimator for the parameters of interest than does using the true values+ This phenomenon takes place even in cases of semi-nonparametric models in which the nuisance parameters are infinitedimensional and cannot be estimated at the parametric rate+ We examine the structure and present the necessary and sufficient condition for the occurrence of this puzzle+ We also provide a simple sufficient condition+ It shows that the puzzle occurs when the term accounting for the effect of estimation of nuisance parameters is included in the tangent space+ This condition is often satisfied when the estimating equation does not bring any restriction on the form of the nuisance parameters+ Our simple sufficient condition can be applied to many important estimators+