Why Well Spread Probability Samples Are Balanced

Affiliation(s)

Department of Forest Resource Management, Swedish University of Agricultural Sciences, Ume?, Sweden.

Department of Mathematics and Mathematical Statistics, Ume? University, Ume?, Sweden.

Department of Forest Resource Management, Swedish University of Agricultural Sciences, Ume?, Sweden.

Department of Mathematics and Mathematical Statistics, Ume? University, Ume?, Sweden.

ABSTRACT

When sampling from a finite population there is often auxiliary information available on unit level. Such information can be used to improve the estimation of the target parameter. We show that probability samples that are well spread in the auxiliary space are balanced, or approximately balanced, on the auxiliary variables. A consequence of this balancing effect is that the Horvitz-Thompson estimator will be a very good estimator for any target variable that can be well approximated by a Lipschitz continuous function of the auxiliary variables. Hence we give a theoretical motivation for use of well spread probability samples. Our conclusions imply that well spread samples, combined with the Horvitz- Thompson estimator, is a good strategy in a varsity of situations.

Cite this paper

A. Grafström and N. Lundström, "Why Well Spread Probability Samples Are Balanced,"*Open Journal of Statistics*, Vol. 3 No. 1, 2013, pp. 36-41. doi: 10.4236/ojs.2013.31005.

A. Grafström and N. Lundström, "Why Well Spread Probability Samples Are Balanced,"

References

[1] L. Barabesi and S. Franceschi, “Sampling Properties of Spatial Total Estimators under Tessellation Stratified Designs,” Environmetrics, Vol. 22, No. 3, 2011, pp. 271- 278. doi:10.1002/env.1046

[2] D. L. Stevens Jr. and A. R. Olsen, “Spatially Balanced Sampling of Natural Resources,” Journal of the American Statistical Association, Vol. 99, No. 465, 2004, pp. 262- 278. doi:10.1198/016214504000000250

[3] J.-C. Deville and Y. Tillé, “Efficient Balanced Sampling: the Cube Method,” Biometrika, Vol. 91, No. 4, 2004, pp. 893-912. doi:10.1093/biomet/91.4.893

[4] J.-C. Deville and Y. Tillé, “Unequal Probability Sampling without Replacement through a Splitting Method,” Biometrika, Vol. 85, No. 1, 1998, pp. 89-101. doi:10.1093/biomet/85.1.89

[5] A. Grafstr?m, N. L. P. Lundstr?m and L. Schelin, “Spatially Balanced Sampling through the Pivotal Method,” Biometrics, Vol. 68 No. 2, 2012, pp. 514-520. doi:10.1111/j.1541-0420.2011.01699.x

[6] A. Grafstr?m, “Spatially Correlated Poisson Sampling,” Journal of Statistical Planning and Inference, Vol. 142, No. 1, 2012, pp. 139-147. doi:10.1016/j.jspi.2011.07.003

[7] L. Bondesson and D. Thorburn, “A List Sequential Sampling Method Suitable for Real-Time Sampling,” Scandinavian Journal of Statistics, Vol. 35, No. 3, 2008, pp. 466-483. doi:10.1111/j.1467-9469.2008.00596.x

[8] D. G. Horvitz and D. J. Thompson, “A Generalization of Sampling without Replacement from a Finite Universe,” Journal of the American Statistical Association, Vol. 47, No. 260, 1952, pp. 663-685. doi:10.1080/01621459.1952.10483446

[9] A. Grafstr?m and Y. Tillé, “Doubly Balanced Spatial Sampling with Spreading and Restitution of Auxiliary Totals,” Environmetrics, in Press, 2012. doi:10.1002/env.2194

[10] A. J. Lister and C. T. Scott, “Use of Space-Filling Curves to Select Sample Locations in Natural Resource Monitoring Studies,” Environmental Monitoring and Assessment, Vol. 149, No. 1-4, 2009, pp. 71-80. doi:10.1007/s10661-008-0184-y

[11] A. Grafstr?m and L. Schelin, “How to Select Representative Samples,” Unpublished, 2012.

[12] D. L. Jr. Stevens and A. R. Olsen, “Variance Estimation for Spatially Balanced Samples of Environmental Resources,” Environmetrics, Vol. 14, No. 6, 2003, pp. 593- 610. doi:10.1002/env.606

[13] N. A. C. Cressie, “Statistics for spatial data,” Wiley, New York, 1993.

[14] L. Barabesi and M. Marcheselli, “A Modified Monte Carlo Integration,” International Mathematical Journal, Vol. 3, No. 5, 2003, pp. 555-565.

[1] L. Barabesi and S. Franceschi, “Sampling Properties of Spatial Total Estimators under Tessellation Stratified Designs,” Environmetrics, Vol. 22, No. 3, 2011, pp. 271- 278. doi:10.1002/env.1046

[2] D. L. Stevens Jr. and A. R. Olsen, “Spatially Balanced Sampling of Natural Resources,” Journal of the American Statistical Association, Vol. 99, No. 465, 2004, pp. 262- 278. doi:10.1198/016214504000000250

[3] J.-C. Deville and Y. Tillé, “Efficient Balanced Sampling: the Cube Method,” Biometrika, Vol. 91, No. 4, 2004, pp. 893-912. doi:10.1093/biomet/91.4.893

[4] J.-C. Deville and Y. Tillé, “Unequal Probability Sampling without Replacement through a Splitting Method,” Biometrika, Vol. 85, No. 1, 1998, pp. 89-101. doi:10.1093/biomet/85.1.89

[5] A. Grafstr?m, N. L. P. Lundstr?m and L. Schelin, “Spatially Balanced Sampling through the Pivotal Method,” Biometrics, Vol. 68 No. 2, 2012, pp. 514-520. doi:10.1111/j.1541-0420.2011.01699.x

[6] A. Grafstr?m, “Spatially Correlated Poisson Sampling,” Journal of Statistical Planning and Inference, Vol. 142, No. 1, 2012, pp. 139-147. doi:10.1016/j.jspi.2011.07.003

[7] L. Bondesson and D. Thorburn, “A List Sequential Sampling Method Suitable for Real-Time Sampling,” Scandinavian Journal of Statistics, Vol. 35, No. 3, 2008, pp. 466-483. doi:10.1111/j.1467-9469.2008.00596.x

[8] D. G. Horvitz and D. J. Thompson, “A Generalization of Sampling without Replacement from a Finite Universe,” Journal of the American Statistical Association, Vol. 47, No. 260, 1952, pp. 663-685. doi:10.1080/01621459.1952.10483446

[9] A. Grafstr?m and Y. Tillé, “Doubly Balanced Spatial Sampling with Spreading and Restitution of Auxiliary Totals,” Environmetrics, in Press, 2012. doi:10.1002/env.2194

[10] A. J. Lister and C. T. Scott, “Use of Space-Filling Curves to Select Sample Locations in Natural Resource Monitoring Studies,” Environmental Monitoring and Assessment, Vol. 149, No. 1-4, 2009, pp. 71-80. doi:10.1007/s10661-008-0184-y

[11] A. Grafstr?m and L. Schelin, “How to Select Representative Samples,” Unpublished, 2012.

[12] D. L. Jr. Stevens and A. R. Olsen, “Variance Estimation for Spatially Balanced Samples of Environmental Resources,” Environmetrics, Vol. 14, No. 6, 2003, pp. 593- 610. doi:10.1002/env.606

[13] N. A. C. Cressie, “Statistics for spatial data,” Wiley, New York, 1993.

[14] L. Barabesi and M. Marcheselli, “A Modified Monte Carlo Integration,” International Mathematical Journal, Vol. 3, No. 5, 2003, pp. 555-565.