On Subsampling Estimators with Unknown Rate of Convergence

Abstract

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 X1,…, Xn. The subsampling methodology hinges on approximating the large-sample distribution of a statistic Tn = Tn(X1,…, Xn) that is consistent for θ at some known rate τn. 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 τn 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 τn, and (b) using the estimated rate to construct asymptotically correct confidence regions for θ based on subsampling. © 1999 Taylor & Francis Group, LLC.