Random Designs for Estimating Integrals of Stochastic Processes

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.