The Annals of Statistics

Random Designs for Estimating Integrals of Stochastic Processes

Carol Schoenfelder and Stamatis Cambanis

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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.

Article information

Ann. Statist., Volume 10, Number 2 (1982), 526-538.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G00
Secondary: 62K05: Optimal designs

Random designs estimation of integrals of stochastic processes simple random sampling stratified sampling systematic sampling


Schoenfelder, Carol; Cambanis, Stamatis. Random Designs for Estimating Integrals of Stochastic Processes. Ann. Statist. 10 (1982), no. 2, 526--538. doi:10.1214/aos/1176345793.

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