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 $K = 0,1,2,\ldots$, 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, $K \geq 0$. 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 $K = 0,1,2,\ldots$.