## The Annals of Statistics

### On optimal spatial subsample size for variance estimation

#### Abstract

We consider the problem of determining the optimal block (or subsample) size for a spatial subsampling method for spatial processes observed on regular grids. We derive expansions for the mean square error of the subsampling variance estimator, which yields an expression for the theoretically optimal block size. The optimal block size is shown to depend in an intricate way on the geometry of the spatial sampling region as well as characteristics of the underlying random field. Final expressions for the optimal block size make use of some nontrivial estimates of lattice point counts in shifts of convex sets. Optimal block sizes are computed for sampling regions of a number of commonly encountered shapes. Numerical studies are performed to compare subsampling methods as well as procedures for estimating the theoretically best block size.

#### Article information

Source
Ann. Statist., Volume 32, Number 5 (2004), 1981-2027.

Dates
First available in Project Euclid: 27 October 2004

https://projecteuclid.org/euclid.aos/1098883779

Digital Object Identifier
doi:10.1214/009053604000000779

Mathematical Reviews number (MathSciNet)
MR2102500

Zentralblatt MATH identifier
1056.62055

Subjects
Primary: 62G09: Resampling methods
Secondary: 62M40: Random fields; image analysis 60G60: Random fields

#### Citation

Nordman, Daniel J.; Lahiri, Soumendra N. On optimal spatial subsample size for variance estimation. Ann. Statist. 32 (2004), no. 5, 1981--2027. doi:10.1214/009053604000000779. https://projecteuclid.org/euclid.aos/1098883779

#### References

• Billingsley, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.
• Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. Ann. Probab. 10 1047--1050.
• Bradley, R. C. (1989). A caution on mixing conditions for random fields. Statist. Probab. Lett. 8 489--491.
• Bühlmann, P. and Künsch, H. R. (1999). Block length selection in the bootstrap for time series. Comput. Statist. Data Anal. 31 295--310.
• Carlstein, E. (1986). The use of subseries values for estimating the variance of a general statistic from a stationary time series. Ann. Statist. 14 1171--1179.
• Chan, G. and Wood, A. T. A. (1997). Algorithm AS 312: An algorithm for simulating stationary Gaussian random fields. Appl. Statist. 46 171--181.
• Cressie, N. (1991). Statistics for Spatial Data. Wiley, New York.
• Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statist. 85. Springer, New York.
• Fukuchi, J.-I. (1999). Subsampling and model selection in time series analysis. Biometrika 86 591--604.
• Garcia-Soidan, P. H. and Hall, P. (1997). On sample reuse methods for spatial data. Biometrics 53 273--281.
• Guyon, X. (1995). Random Fields on a Network: Modelling, Statistics and Applications. Springer, New York.
• Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
• Hall, P., Horowitz, J. L. and Jing, B.-Y. (1995). On blocking rules for the bootstrap with dependent data. Biometrika 82 561--574.
• Hall, P. and Jing, B.-Y. (1996). On sample reuse methods for dependent data. J. Roy. Statist. Soc. Ser. B 58 727--737.
• Huxley, M. N. (1993). Exponential sums and lattice points. II. Proc. London Math. Soc. (3) 66 279--301.
• Huxley, M. N. (1996). Area, Lattice Points, and Exponential Sums. Oxford Univ. Press, New York.
• Krätzel, E. (1988). Lattice Points. Deutscher Verlag Wiss., Berlin.
• Künsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. Ann. Statist. 17 1217--1241.
• Lahiri, S. N. (1996). On empirical choice of the optimal block size for block bootstrap methods. Preprint, Dept. Statistics, Iowa State Univ.
• Lahiri, S. N. (1999a). Asymptotic distribution of the empirical spatial cumulative distribution function predictor and prediction bands based on a subsampling method. Probab. Theory Related Fields 114 55--84.
• Lahiri, S. N. (1999b). Theoretical comparisons of block bootstrap methods. Ann. Statist. 27 386--404.
• Lahiri, S. N. (2004). Central limit theorems for weighted sums of a spatial process under a class of stochastic and fixed designs. Sankhyā. To appear.
• Lahiri, S. N., Furukawa, K. and Lee, Y.-D. (2003). A nonparametric plug-in rule for selecting optimal block lengths for block bootstrap methods. Preprint, Dept. Statistics, Iowa State Univ.
• Léger, C., Politis, D. N. and Romano, J. P. (1992). Bootstrap technology and applications. Technometrics 34 378--399.
• Martin, R. J. (1990). The use of time-series models and methods in the analysis of agricultural field trials. Comm. Statist. Theory Methods 19 55--81.
• Meketon, M. S. and Schmeiser, B. (1984). Overlapping batch means: Something for nothing? In Proc. 16th Conference on Winter Simulation Conf. (S. Sheppard, U. Pooch and D. Pegden, eds.) 227--230. IEEE, Piscataway, NJ.
• Nordman, D. J. (2002). On optimal spatial subsample size for variance estimation. Ph.D. dissertation, Dept. Statistics, Iowa State Univ.
• Nordman, D. J. and Lahiri, S. N. (2002). On the approximation of differenced lattice point counts with application to statistical bias expansions. Preprint, Dept. Statistics, Iowa State Univ.
• Nordman, D. J. and Lahiri, S. N. (2003). On optimal variance estimation under different spatial subsampling schemes. In Recent Advances and Trends in Nonparametric Statistics (M. G. Akritas and D. N. Politis, eds.). North-Holland, Amsterdam.
• Perera, G. (1997). Geometry of $\mathbbZ^d$ and the central limit theorem for weakly dependent random fields. J. Theoret. Probab. 10 581--603.
• Politis, D. N. and Romano, J. P. (1993a). Nonparametric resampling for homogeneous strong mixing random fields. J. Multivariate Anal. 47 301--328.
• Politis, D. N. and Romano, J. P. (1993b). On the sample variance of linear statistics derived from mixing sequences. Stochastic Process. Appl. 45 155--167.
• Politis, D. N. and Romano, J. P. (1994). Large sample confidence regions based on subsamples under minimal assumptions. Ann. Statist. 22 2031--2050.
• Politis, D. N. and Romano, J. P. (1995). Bias-corrected nonparametric spectral estimation. J. Time Ser. Anal. 16 67--103.
• Politis, D. N., Romano, J. P. and Wolf, M. (1999). Subsampling. Springer, New York.
• Politis, D. N. and Sherman, M. (2001). Moment estimation for statistics from marked point processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 261--275.
• Possolo, A. (1991). Subsampling a random field. In Spatial Statistics and Imaging (A. Possolo, ed.) 286--294. IMS, Hayward, CA.
• Ripley, B. D. (1981). Spatial Statistics. Wiley, New York.
• Sherman, M. (1996). Variance estimation for statistics computed from spatial lattice data. J. Roy. Statist. Soc. Ser. B 58 509--523.
• Sherman, M. and Carlstein, E. (1994). Nonparametric estimation of the moments of a general statistic computed from spatial data. J. Amer. Statist. Assoc. 89 496--500.
• van der Corput, J. G. (1920). Über Gitterpunkte in der Ebene. Math. Ann. 81 1--20.