Bootstrap methods for dependent data: A review

This paper gives a review on a variety of bootstrap methods for dependent data. The main focus is not on an exhaustive listing and description of bootstrap procedures but on general principles which should be taken into account when selecting a particular bootstrap procedure in order to approximate the (properly standardized) distribution of a statistic of interest. Questions are considered related to which dependence properties of the underlying data generating process asymptotically influence the distribution of the statistic of interest and which dependence properties (or even which process) a particular bootstrap method really mimics. For answering these questions we introduce the concept of a companion stochastic process. As statistics we consider generalized means, and integrated periodogram statistics (including ratio statistics) as well as nonparametric estimators.