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Model-based variance estimation in non-measurable spatial designs. (English) Zbl 1356.62224

Summary: Two-dimensional systematic sampling and maximal stratification are frequently used in spatial surveys, because of their ease of implementation and design efficiency. An important drawback of these designs, however, is that no direct estimator of the design variance is available. In this paper estimation of the sampling variance of a total in a model-based context is considered. { }The estimation strategy is based on the use of the sample variogram which can be either a non-parametric or a parametric one. Consistency of the estimators is discussed; simulations and an application to real data show the good performance of the proposed procedure in practice.

MSC:

62P12 Applications of statistics to environmental and related topics
62D05 Sampling theory, sample surveys
62M30 Inference from spatial processes
62H12 Estimation in multivariate analysis

Software:

spatial
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References:

[1] Bartolucci, F.; Montanari, G. E., A new class of unbiased estimators of the variance of the systematic sample mean, J. Statist. Plann. Inference, 136, 4, 1512-1525 (2006) · Zbl 1088.62013
[2] (Benedetti, R.; Piersimoni, F.; Bee, M.; Espa, G., Agricultural Survey Methods (2010), Wiley: Wiley New York)
[3] Benedetti, R.; Piersimoni, F.; Postiglione, P., Sampling Spatial Units for Agricultural Surveys (2015), Springer: Springer Berlin
[4] Besag, J., Spatial interaction and the statistical analysis of lattice systems, J. R. Stat. Soc. Ser. B Stat. Methodol., 36, 192-236 (1974) · Zbl 0327.60067
[5] Cliff, A. D.; Ord, J. K., Spatial Processes: Models and Applications (1981), Wiley: Wiley New York · Zbl 0353.62088
[6] Cochran, W. G., Sampling Techniques (1977), Pion: Pion London · Zbl 0051.10707
[7] Cressie, N. A.C., Statistics for Spatial Data (1993), John Wiley & Sons: John Wiley & Sons New Jersey · Zbl 0468.62095
[8] Dickson, M. M.; Benedetti, R.; Giuliani, D.; Espa, G., The use of spatial sampling designs in business surveys, Open J. Stat., 4, 345-354 (2014)
[9] Diggle, P. J.; Ribeiro, P. J., Model-Based Geostatistics (2007), Springer: Springer New York · Zbl 1132.86002
[10] D’Orazio, M., Estimating the variance of the sample mean in two-dimensional systematic sampling, J. Agric. Biol. Environ. Stat., 8, 280-295 (2003)
[11] Fernández-Casal, R.; Francisco-Fernández, M., Nonparametric bias-corrected variogram estimation under non-constant trend, Stoch. Environ. Res. Risk Assess., 28, 1247-1259 (2014)
[12] Fernández Casal, R.; González Manteiga, W.; Febrero Bande, M., Space-time dependency modeling using general classes of flexible stationary variogram models, J. Geophys. Res., 108, 8779 (2003)
[13] Fewster, R. M., Variance estimation for systematic designs in spatial surveys, Biometrics, 67, 1518-1531 (2011) · Zbl 1274.62770
[14] Fuller, W. A., Sampling Statistics (2009), John Wiley & Sons: John Wiley & Sons New Jersey · Zbl 1179.62019
[15] Garcia-Soidan, P. H.; González-Manteiga, W.; Febrero-Bande, M., Local linear regression estimation of the variogram, Statist. Probab. Lett., 64, 169-179 (2003) · Zbl 1113.62318
[16] Goovaerts, P., Geostatistics for Natural Resources Evaluation (1997), Oxford University Press on Demand
[17] Grafström, A.; Tillé, Y., Doubly balanced spatial sampling with spreading and restitution of auxiliary totals, Environmetrics, 24, 120-131 (2013)
[18] Gregoire, T. G.; Valentine, H. T., Sampling Strategies for Natural Resources and the Environment (2007), Chapman & Hall: Chapman & Hall New York
[19] Hawkins, D. M.; Cressie, N. A.C., Robust kriging-a proposal, J. Int. Ass. Math. Geol., 16, 3-18 (1984)
[20] Journel, A. G.; Huijbregts, C. J., Mining Geostatistics (1978), Academic Press: Academic Press London
[21] Knautz, H.; Trenkler, G., Bounds for bias and variance of \(S^2\) under dependence, Scand. J. Statist., 22, 121-128 (1995) · Zbl 0819.62019
[22] Leonenko, N.; Taufer, E., Disaggregation of spatial autoregressive processes, Spat. Stat., 3, 1-20 (2013)
[23] Mardia, K. V.; Marshall, R. J., Maximum likelihood estimation of models for residual covariance in spatial regression, Biometrika, 71, 135-146 (1984) · Zbl 0542.62079
[24] Matheron, G., (The Theory of Regionalised Variables and its Applications. The Theory of Regionalised Variables and its Applications, École National Supérieure des Mines, vol. 5 (1971))
[25] Mercer, W. B.; Hall, A. D., The experimental error of field trials, J. Agric. Sci., 4, 107-132 (1911)
[26] Opsomer, J. D.; FranciscoFernández, M.; Xiaoxi, Li., Modelbased nonparametric variance estimation for systematic sampling, Scand. J. Statist., 39, 528-542 (2012) · Zbl 1323.62015
[27] Patankar, V. N., The goodness of fit of the frequency distribution obtained from stochastic processes, Biometrika, 41, 450-462 (1954) · Zbl 0056.37202
[28] Ripley, B. D., Spatial Statistics (1981), John Wiley & Sons: John Wiley & Sons New York · Zbl 0558.62083
[29] Särndal, C. E.; Swensson, B.; Wretman, J., Model Assisted Survey Sampling (1992), Springer: Springer New York · Zbl 0742.62008
[30] Stevens, D. L.; Olsen, A. R., Spatially balanced sampling of natural resources, J. Amer. Statist. Assoc., 99, 262-278 (2004) · Zbl 1092.62500
[31] Tan, Kim H., Soil Sampling, Preparation, and Analysis (2005), CRC Press
[32] Tobler, W., A computer movie simulating urban growth in the Detroit region, Econ. Geogr., 46, 234-240 (1970)
[33] Wang, J. F.; Christakos, G.; Hu, M. G., Modeling spatial means of surfaces with stratified nonhomogeneity, IEEE Trans. Geosci. Remote Sens., 47, 12, 4167-4174 (2009)
[34] Wang, J.; Haining, R.; Cao, Z., Sample surveying to estimate the mean of a heterogeneous surface: reducing the error variance through zoning, Int. J. Geogr. Inf. Sci., 24, 4, 523-543 (2010)
[35] Wang, J. F.; Haining, R.; Liu, T. J.; Li, L. F.; Jiang, C. S., Sandwich estimation for multi-unit reporting on a stratified heterogeneous surface, Environ. Plann. A, 45, 10, 2515-2534 (2013)
[36] Wang, J. F.; Stein, A.; Gao, B. B.; Ge, Y., A review of spatial sampling, Spat. Stat., 2, 1-14 (2012)
[37] Wang, J. F.; Zhang, T. L.; Fu, B. J., A measure of spatial stratified heterogeneity, Ecol. Indic., 67, 250-256 (2016)
[38] Whittle, P., On stationary processes in the plane, Biometrika, 41, 434-449 (1954) · Zbl 0058.35601
[39] Wolter, K. M., Introduction to Variance Estimation (2007), Springer-Verlag Inc.: Springer-Verlag Inc. New York · Zbl 1284.62023
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