## 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 $$K=0,1,2,\dots$$.

### MSC:

 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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