Comparison of Block Bootstrap Methods

Abstract

In this chapter, we compare the performance of the MBB, the NBB, the CBB, and the SB methods considered in Chapters 3 and 4. In Section 5.2, we present a simulated data example and illustrate the behavior of the block bootstrap methods under some simple time series models. Although the example treats the simple case of the sample mean, it provides a representative picture of the properties of the four methods in more general problems. In the subsequent sections, the empirical findings of Section 5.2 are substantiated through theoretical results that provide a comparison of the methods in terms of the (asymptotic) MSEs of the bootstrap estimators. In Section 5.3, we describe the framework for the theoretical comparison. In Section 5.4, we obtain expansions for the MSEs of the relevant bootstrap estimators as a function of the block size (expected block size, for the SB). These expansions provide the basis for the theoretical comparison of the sampling properties of the bootstrap methods. In Section 5.5, the main theoretical findings are presented. Here, we compare the bootstrap methods using the leading terms in the expansions of the MSEs derived in the previous section. In Section 5.5, we also derive theoretical optimal (expected) block lengths for each of the block bootstrap estimators and compare the methods at the corresponding optimal block lengths. Some conclusions and implications of the theoretical and finite sample simulation results are discussed in Section 5.6. Proofs of two key results from Section 5.4 are separated out into Section 5.7.