Edgeworth approximation in semiparametric regression models

The properties of a class of semiparametric estimators are examined using Edgeworth approximation methods. These approximations provide useful information which is not available from the first order asymptotic theory. We use the approximations to compare alternative estimators that are first order equivalent. We also develop an optimality theory based on the second order approximations, and suggest how to construct estimators that are second order optimal.;Chapter 4 calibrates some of the formulae presented in Chapter 3 and describes the results of a few monte carlo experiments. Finally, in Chapter 5 we discuss distributional approximation and prove a theorem that justifies the Edgeworth approximation for a class of weighted U statistics on which our estimators depend.;In Chapter 1 we introduce the class of regression models we shall examine. In Chapter 2 we describe methods of constructing semiparametric estimators that have desirable first order properties. The main goal in this chapter is to demonstrate that such estimators can be approximated by certain polynomial statistics whose properties are easier to evaluate. In Chapter 3 we consider in detail these polynomial statistics. We construct second order approximations to the first two moments for a number of the examples considered in Chapter 1. These moments reflect certain features of the estimation method that are not present in the first order theory. In particular, they depend on the choice of bandwidth and weighting function, and on the smoothness of the nuisance parameters. These approximations are then used to describe second order optimal procedures that can be implemented.