Extensions and Examples

Abstract

In this chapter, we establish consistency of different block bootstrap methods for some general classes of estimators and consider some specific examples illustrating the theoretical results. Section 4.2 establishes consistency of estimators that may be represented as smooth functions of sample means. Section 4.3 deals with (generalized) M-estimators, including the maximum likelihood estimators of parameters, which are defined through estimating equations. Some special considerations are required while defining the bootstrap versions of such estimators. We describe the relevant issues in detail in Section 4.3. Section 4.4 gives results on the bootstrapped empirical process, and establishes consistency of bootstrap estimators for certain differentiable statistical functionals. Section 4.5 contains three numerical examples, illustrating the theoretical results of Sections 4.2–4.4.