A note on the stationary bootstrap's variance

TL;DR

This paper reveals that the stationary bootstrap's variance is comparable to nonoverlapping block bootstrap, offering a new perspective and a unified method for variance analysis in block bootstrap techniques.

Contribution

It demonstrates that the stationary bootstrap's variance matches that of nonoverlapping block bootstrap, providing a frequency domain approach for variance determination.

Findings

Stationary bootstrap variance equals nonoverlapping block bootstrap variance.

Unified frequency domain method for variance calculation.

Updated results on efficiency and optimal block size.

Abstract

Because the stationary bootstrap resamples data blocks of random length, this method has been thought to have the largest asymptotic variance among block bootstraps Lahiri [Ann. Statist. 27 (1999) 386--404]. It is shown here that the variance of the stationary bootstrap surprisingly matches that of a block bootstrap based on nonrandom, nonoverlapping blocks. This argument translates the variance expansion into the frequency domain and provides a unified way of determining variances for other block bootstraps. Some previous results on the stationary bootstrap, related to asymptotic relative efficiency and optimal block size, are also updated.