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March, 1992 Sampling Designs for Estimating Integrals of Stochastic Processes
Karim Benhenni, Stamatis Cambanis
Ann. Statist. 20(1): 161-194 (March, 1992). DOI: 10.1214/aos/1176348517


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


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Karim Benhenni. Stamatis Cambanis. "Sampling Designs for Estimating Integrals of Stochastic Processes." Ann. Statist. 20 (1) 161 - 194, March, 1992.


Published: March, 1992
First available in Project Euclid: 12 April 2007

zbMATH: 0749.60033
MathSciNet: MR1150339
Digital Object Identifier: 10.1214/aos/1176348517

Primary: 60G12
Secondary: 62K05 , 62M99 , 65B15 , 65D30

Keywords: integral approximation , regular sampling designs , Second-order process , weighted Euler-MacLaurin and Gregory formulae

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 1 • March, 1992
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