THE STATIONARY BOOTSTRAP

This article introduces a resampling procedure called the stationary b ootstrap as a means of calculating standard errors of estimators and c onstructing confidence regions for parameters based on weakly dependen t stationary observations. Previously, a technique based on resampling blocks of consecutive observations was introduced to construct confid ence intervals for a parameter of the m-dimensional joint distribution of m consecutive observations, where m is fixed. This procedure has b een generalized by constructing a ''blocks of blocks'' resampling sche me that yields asymptotically valid procedures even for a multivariate parameter of the whole (i.e., infinite-dimensional) joint distributio n of the stationary sequence of observations. These methods share the construction of resampling blocks of observations to form a pseudo-tim e series, so that the statistic of interest may be recalculated based on the resampled data set. But in the context of applying this method to stationary data, it is natural to require the resampled pseudo-time series to be stationary (conditional on the original data) as well. A lthough the aforementioned procedures lack this property, the stationa ry procedure developed here is indeed stationary and possesses other d esirable properties. The stationary procedure is based on resampling b locks of random length, where the length of each block has a geometric distribution. In this article, fundamental consistency and weak conve rgence properties of the stationary resampling scheme are developed.