SAMPLING DESIGNS FOR ESTIMATING INTEGRALS OF STOCHASTIC-PROCESSES

The problem of estimating the integral of a stochastic process from observations at a finite number of sampling points is considered. Sacks and Ylvisaker found a sequence of asymptotically optimal sampling designs for general processes with exactly 0 and 1 quadratic mean (q.m.) derivatives using optimal-coefficient estimators, which depend on the process covariance. These results were extended to a restricted class of processes with exactly K q.m. derivatives, for all K = 0, 1, 2,..., by Eubank, Smith and Smith. The asymptotic performance of these optimal-coefficient estimators is determined here for regular sequences of sampling designs and general processes with exactly K q.m. derivatives, K greater-than-or-equal-to 0. More significantly, simple nonparametric estimators based on an adjusted trapezoidal rule using regular sampling designs are introduced whose asymptotic performance is identical to that of the optimal-coefficient estimators for general processes with exactly K q.m. derivatives for all K = 0, 1, 2,....