Dependent Data Bootstrap

# Dependent Data Bootstrap - Bootstrapping Dependent Data One...

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Bootstrapping Dependent Data One of the key issues confronting bootstrap resampling approximations is how to deal with dependent data. Consider a sequence f X t g n t =1 of dependent random variables. Clearly it would be a mistake to resample from the sequence scalar quantities, as the reshu› ed resamples would break the temporal dependence. Our goal is most often to learn the variance of a general statistic T n ( X 1 ;:::;X n ) , we hereafter refer to the unknown variance as 2 . The quantity 2 may not be calculable analytically because the dependence structure and the underlying distribution of the innovations are not assumed to be known. In 1985, Hall examined the problem of bootstrap estimation for data that was spatial in character. His proposed methods could be applied to time-series 1 For m nonoverlapping blocks of equal length, each block has length n m . For the moving-block bootstrap, he proposes dividing the series into n m +1 overlapping blocks of equal length n m . f x 1 ;:::;x 4 g bootstrap is obtained by constructing the statistic of interest for each member of the set f ( x 1 ;x 2 ) ; ( x 3 ;x 4 ) g : The moving-block bootstrap is obtained by constructing the statistic of interest for each member of the set f ( x 1 ;x 2 ) ; ( x 2 ;x 3 ) ; ( x 3 ;x 4 ) g : The intuition underpinning the &xed-block bootstrap is as follows. The moving- block bootstrap has many samples that share a large number of observations, in Further, if m grows with n , then a statistic constructed from a given subsample will eventually behave as though it is independent of all but two (the adjacent two) of the statistics constructed from the other subsamples. In addition, m should grow with n to allow for long-lived dynamics to be captured. One natural choice for m would be m = cn , with 0 < c < 1 , as the subsamples would be of the same order of magnitude as the original data. Unfortunately, such an approach would 1 To see why consider the case of two dimensional spatial data. Rather than a sequence of

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Dependent Data Bootstrap - Bootstrapping Dependent Data One...

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