Efficient Inference for Finite-Dimensional Parameters

In this chapter, as in Section 3.2, we focus on semiparametric models {P _θ,η : θ ∈ θ, η ∈ H}, where θ is an open subset of R^k and H is an arbitrary, possibly infinite-dimensional set. The parameter of interest for this chapter is ψ(P _θ,η ) = θ.

We first present the promised proof of Theorem 3.1. It may at first appear that Theorem 3.1 can only be used when the form of the efficient score equation can be written down explicitly. However, even in those settings where the efficient score cannot be written in closed form, Theorem 3.1 and a number of related approaches can be used for semiparametric efficient estimation based on the profile likelihood. This process can be facilitated through the use of approximately least-favorable submodels, which are discussed in the second section.

The main ideas of this chapter are given in Section 19.3, which presents several methods of inference for θ that go significantly beyond Theorem 3.1. The first two methods are based on a multivariate normal approximation of the profile likelihood that is valid in a neighborhood of the true parameter. The first of the two methods is based on a quadratic expansion that is valid in a shrinking neighborhood. The second method, the profile sampler is based on an expansion that is valid on a compact, fixed neighborhood. The third method is an extension that is valid for penalized profile likelihoods. A few other methods are also discussed, including bootstrap, jackknife, and fully Bayesian approaches.