Sampling designs for estimating integrals of stochastic processes. (English) Zbl 0749.60033

Summary: The problem of estimating the integral of a stochastic process from observations at a finite number of sampling points is considered. J. Sacks and D. Ylvisaker [Ann. Math. Stat. 37, 66-89 (1966; Zbl 0152.175); ibid. 39, 49-69 (1968; Zbl 0165.215), ibid. 41, 2057-2074 (1970; Zbl 0234.62025), and Time ser. stoch. Processes, Convexity Combinat., Proc. 12th bien. Sem. Canadian math. Congr. 1969, 115-136 (1970; Zbl 0291.62090)] 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 \(K=0,1,2,\dots\), by R. L. Eubank, P. L. Smith and P. W. Smith [Ann. Stat. 10, 1295-1301 (1982; Zbl 0522.62055)]. 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 \(K=0,1,2,\dots\).


60G12 General second-order stochastic processes
65D30 Numerical integration
62K05 Optimal statistical designs
62M99 Inference from stochastic processes
65B15 Euler-Maclaurin formula in numerical analysis
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