Asymptotic optimality of regular sequence designs

Abstract

We study linear estimators for the weighted integral of a stochastic process. The process may only be observed on a finite sampling design. The error is defined in a mean square sense, and the process is assumed to satisfy Sacks-Ylvisaker regularity conditions of order $r \epsilon \mathbb{N}_0$. We show that sampling at the quantiles of a particular density already yields asymptotically optimal estimators. Hereby we extend the results of Sacks and Ylvisaker for regularity $r = 0$ or 1, and we confirm a conjecture by Eubank, Smith and Smith.