Statistica Sinica

Dimitris N. Politis and Joseph P. Romano

Abstract:In this article, some weak convergence results are developed for approximate sums of weakly dependent stationary Hilbert space valued random variables in a triangular array setting. The motivation for such results lies in understanding the weak convergence properties of estimators which are smooth functionals of the empirical process. By regarding the empirical process as an element of an appropriate Hilbert space, the asymptotic distributional properties can be deduced. The results are designed to be strong enough to handle the study of estimators under the stationary bootstrap resampling plan. In Politis and Romano (1991), a resampling procedure, called the stationary bootstrap, is introduced as a means of calculating standard errors of estimators and constructing confidence regions for parameters based on weakly dependent stationary observations. The results derived here support the asymptotic validity of the stationary bootstrap method for a broad class of estimators. In particular, minimum distance estimators, whose robustness properties have been well-established by Millar (1981, 1984), are shown to have robustness of validity in the sense that confidence intervals constructed by the stationary bootstrap method based on such estimators are asymptotically valid even when the usual independence assumption often used in robustness studies is seriously violated.

Key words and phrases:Approximate confidence limit, bootstrap, differentiability, Hilbert space, minimum distance estimators, stationary, time series.