Geodesic
Adjusted Euler–MacLaurin predictor for
Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 3, pp. 596-608 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the problem of predicting integrals of a spatial stationary process $Z$ over a unit square. We construct predictors based on a systematic sampling of size $m^2$ by approximating the existing mean squared derivatives of the process in the two-dimensional Euler–MacLaurin formula by finite differences up to some appropriate order. We show that if the spectral density satisfies $f_{Z}(\omega) =o(|\omega|^{-p})$ for any fixed positive integer $p$, the corresponding mean squared error is of order $m^{-p}$.
Keywords: spatial stationary process, predictor, Euler–MacLaurin formula. 
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K. Benhenni; R. Drouilhet. Adjusted Euler–MacLaurin predictor for. Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 3, pp. 596-608. http://geodesic.mathdoc.fr/item/TVP_2003_48_3_a10/

[1] Benhenni K., “Approximating integrals of stochastic processes: Extensions”, J. Appl. Probab., 35:4 (1998), 843–855 | DOI | MR | Zbl

[2] Benhenni K., Cambanis S., “Sampling designs for estimating integrals of stochastic processes”, Ann. Statist., 20:1 (1992), 161–194 | DOI | MR | Zbl

[3] Benhenni K., Cambanis S., “Sampling designs for estimating integrals of stochastic processes using quadratic mean derivatives”, Approximation Theory, ed. G. Anastassiou, Dekker, New York, 1992, 93–128 | MR

[4] Cressie N., Statistics for Spatial Data, John Wiley, New York, 1991 | MR | Zbl

[5] Eubank R. L., Smith P. L., Smith P. W., “A note on optimal and asymptotically optimal designs for certain time series models”, Ann. Statist., 10:4 (1982), 1295–1301 | DOI | MR | Zbl

[6] Istas J., Laredo C., “Estimating functionals of a stochastic process”, Adv. Appl. Probab., 29 (1997), 249–270 | DOI | MR | Zbl

[7] Journel A. G., “New ways of assessing spatial distribution of pollutants”, Environmental Sampling for Hazardous Wastes, eds. G. E. Schweitzer, J. A. Santolucito, American Chemical Society, Washington, D.C., 1984, 109–118

[8] Matern B., Spatial Variation, Lectures Notes in Statist., 36, 2nd ed., Springer, New York, 1986 | MR | Zbl

[9] Olea R. A., “Sampling design optimization for spatial functions”, Math. Geology, 16 (1984), 369–392 | DOI

[10] Ripley B. D., Spatial Statistics, John Wiley, New York, 1981 | MR | Zbl

[11] Ritter K., “Asymptotic optimality of regular sequence designs”, Ann. Statist., 24:5 (1996), 2081–2096 | DOI | MR | Zbl

[12] Sacks J., Ylvisaker D., “Designs for regression problems with correlated errors”, Ann. Math. Statist., 37 (1966), 68–89 | DOI | MR

[13] Sacks J., Ylvisaker D., “Designs for regression problems with correlated errors. III”, Ann. Math. Statist., 41 (1970), 2057–2074 | DOI | MR | Zbl

[14] Sacks J., Ylvisaker D., “Statistical designs and integral approximations”, Proc. Twelfth Biennial Sem. Can. Math. Congr. On Time Series and Stochastic Proceses; Convexity and Combinatorics (Vancouver, B.C., 1969), Montreal, Que., 1970, 115–136 | MR | Zbl

[15] Stein M. L., “Asymptotic properties of centered systematic sampling for predicting integrals of spatial processes”, Ann. Appl. Probab., 3:3 (1993), 874–880 | DOI | MR | Zbl

[16] Stein M. L., “Predicting integrals of stochastic processes”, Ann. Appl. Probab., 5 (1995), 158–170 | DOI | MR | Zbl

[17] Tubilla A., Error convergence rates for estimates of multidimensional integrals of random functions, Tech. Rep., 72, Department of Statistics, Stanford University, 1975

[18] Yfantis E. A., Flatman G. T., Baher J. V., “Efficiency of kriging estimations for square, triangular and hexagonal grids”, J. Math. Geology, 19 (1987), 183–205 | DOI