Application of convolution theorems in semiparametric models with non-i.i.d. data

Abstract

A useful approach to asymptotic efficiency for estimators in semiparametric models is the study of lower bounds on asymptotic variances via convolution theorems. Such theorems are often applicable in models in which the classical assumptions of independence and identical distributions fail to hold, but to date, much of the research has focused on semiparametric models with independent and identically distributed (i.i.d.) data because tools are available in the i.i.d. setting for verifying pre-conditions of the convolution theorems. We develop tools for non-i.i.d. data that are similar in spirit to those for i.i.d. data and also analogous to the approaches used in parametric models with dependent data. This involves extending the notion of the tangent vector figuring so prominently in the i.i.d. theory and providing conditions for smoothness, or differentiability, of the parameter of interest as a function of the underlying probability measures. As a corollary to the differentiability result we obtain sufficient conditions for equivalence, in terms of asymptotic variance bounds, of two models. Regularity and asymptotic linearity of estimators are also discussed.