On subsampling estimators with unknown rate of convergence

Politis and Romano have put forth a general subsampling methodology for the construction of large-sample confidence regions for a general unknown parameter a associated with the probability distribution generating the stationary sequence X-1,...,X-n. The subsampling methodology hinges on approximating the large-sample distribution of a statistic T-n = T-n(X-1,X-...,X-n) that is consistent for theta at some known rate tau(n). Although subsampling has been shown to yield confidence regions for theta of asymptotically correct coverage under very weak assumptions, the applicability of the methodology as it has been presented so far is limited if the rate of convergence tau(n) happens to be unknown or intractable in a particular setting. In this article we show how it is possible to circumvent this limitation by (a) using the subsampling methodology to derive a consistent estimator of the rate tau(n), and (b) using the estimated rate to construct asymptotically correct confidence regions for a based on subsampling.