Introduction

Semi parametric models are statistical models in which the parameter is not a Euclidean vector but ranges over an “infinite-dimensional” parameter set. A different name is “model with a large parameter space.” In the situation in which the observations consist of a random sample from a common distribution P, the model is simply the set P of all possible values of P: a collection of probability measures on the sample space. The simplest type of infinite-dimensional model is the non parametric model, in which we observe a random sample from a completely unknown distribution. Then P is the collection of all probability measures on the sample space, and, as we shall see and as is intuitively clear, the empirical distribution is an asymptotically efficient estimator for the underlying distribution. More interesting are the intermediate models, which are not “nicely” parametrized by a Euclidean parameter, as are the standard classical models, but do restrict the distribution in an important way. Such models are often parametrized by infinite-dimensional parameters, such as distribution functions or densities, that express the structure under study. Many aspects of these parameters are estimable by the same order of accuracy as classical parameters, and efficient estimators are asymptotically normal. In particular, the model may have a natural parametrization. is a Euclidean parameter and 1] runs through a nonparametric class of distributions, or some other infinite-dimensional set. This gives a semiparametric model in the strict sense, in which we aim at estimating and consider 1] as a nuisance parameter. More generally, we focus on estimating the value of some function on the model.

In this chapter we extend the theory of asymptotic efficiency, as developed in Chapters 8 and 15, from parametric to semiparametric models and discuss some methods of estimation and testing. Although the efficiency theory (lower bounds) is fairly complete, there are still important holes in the estimation theory. In particular, the extent to which the lower bounds are sharp is unclear.