blockbootstrap

Block bootstrap is a resampling method used for dependent data, particularly time series, to approximate the sampling distribution of a statistic. By resampling blocks of consecutive observations rather than individual observations, it preserves some of the intrinsic dependence structure that would be broken by naive bootstrap methods.

Implementation typically starts with selecting a block length. In non-overlapping block bootstrap, the data are divided into consecutive blocks of fixed length, and blocks are drawn with replacement to build a bootstrap sample of the original size. In moving block bootstrap, overlapping blocks are formed from all possible consecutive l-tuples, increasing resample variability. Other variants include stationary bootstrap, where block lengths follow a geometrical distribution to model varying dependence, and circular bootstrap, which wraps the series around to form blocks. Tapered or other block constructions are also used in some contexts.

The choice of block length is crucial. Short blocks may fail to capture dependence, while long blocks increase variance and reduce the number of effective resamples. Data-driven methods exist to select l, including plug-in approaches and criteria based on minimizing mean squared error for the statistic of interest, as well as cross-validation techniques.

Block bootstrap is widely used to estimate standard errors and construct confidence intervals for means, autocovariances, regression coefficients, and other statistics in fields such as finance, econometrics, climatology, and engineering. Limitations include sensitivity to nonstationarity, long-range dependence, and heavy tails, as well as increased computational complexity.