Page 67 - Textos de Matemática Vol. 47

BOOTSTRAP AND JACKKNIFE METHODS IN EXTREMAL INDEX ESTIMATION 57
The motivation for this scheme is to preserve the dependence structure of the underlying model within each block. Several ways of blocking have been proposed: Nonoverlapping Block Bootstrap (NBB), Carlstein (1986); Moving Block Bootstrap (MBB), Ku¨nsch, (1989), Liu and Singh, (1992); Circular Block Bootstrap (CBB), Politis and Romano (1992) and Stationary Bootstrap (SB), Politis and Romano (1994).
For each way of blocking it is necessary to consider a length b ⌘ b(n) to resample blocks of observations, but the accuracy of block bootstrap estimators depends critically on the block size for resampling, that must be supplied by the user, (Lahiri et. al., 2007).
Figure 5 illustrates one sample path of size n = 100 generated from the ARMAX process with ✓ = 0.1 and also sample paths obtained from one classical bootstrap resampling and block bootstrap resampling using several block sizes. It is clear that the block size a↵ects strongly the pattern of the resampled values.
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Figure 5. Values from a sample of size n = 100 generated from the ARMAX process with ✓ = 0.1 (top left) and resampled equal size samples considering the classical i.i.d. bootstrap (top right) and blocks of size 5 and 15 (bottom left and right, respectively).
Several authors such as Hall et al. (1995), Bu¨hlman and Ku¨nsch (1999), Politis and White (2004) and Lahiri et al. (2007) proposed ways of estimating the optimal block size.
Here we follow Lahiri et al. (2007), who proposed a nonparametric plug-in (NPPI) method for the empirical choice of the optimal block size for the block