Blockwise bootstrap of the estimated empirical process based on $\psi$-weakly dependent observations

Abstract

The distributional convergence of the bootstrapped estimated empirical process is shown and bootstrap consistency in the $\sup$-norm for test statistics based on that process. Bootstrapping the estimated empirical process has up to now been considered by assuming independence of the observations, where we give up this assumption now and allow the observations to be $\psi$-weakly dependent in the sense of Doukhan and Louhichi (Stoch Proc Appl 84:313–342, 1999). Due to the fact that no model assumptions on the original process are made, a nonparametric blockwise bootstrap procedure is used, which has previously been used in empirical process theory based on mixing observations. Here, we succeeded in proving that assuming $l=o(n)$ and $l\rightarrow \infty$ as only conditions for the blocklength is sufficient to show convergence of the bootstrap process to the same limit as for the original process under ${\mathcal {H}}_0$, which is the weakest condition that has been imposed in that context up to now.