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 θ 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 n ) that is consistent for θ at some known rate τ n . Although subsampling has been shown to yield confidence regions for θ 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 τ 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 τ n , and (b) using the estimated rate to construct asymptotically correct confidence regions for θ based on subsampling.