Random Designs for Estimating Integrals of Stochastic Processes

Abstract

The integral of a second-order stochastic process $Z$ over a $d$-dimensional domain is estimated by a weighted linear combination of observations of $Z$ in a random design. The design sample points are possibly dependent random variables and are independent of the process $Z$, which may be nonstationary. Necessary and sufficient conditions are obtained for the mean squared error of a random design estimator to converge to zero as the sample size increases towards infinity. Simple random, stratified and systematic sampling designs are considered; an optimal simple random design is obtained for fixed sample size; and the mean squared errors of the estimators from these designs are compared. It is shown, for example, that for any simple random design there is always a better stratified design.