Abstract
When estimating the mean value of a variable, or the total amount of a resource, within a specified region it is desirable to report an estimated standard error for the resulting estimate. If the sample sites are selected according to a probability sampling design, it usually is possible to construct an appropriate design-based standard error estimate. One exception is systematic sampling for which no such standard error estimator exists. However, a slight modification of systematic sampling, termed 2-step tessellation stratified (2TS) sampling, does permit the estimation of design-based standard errors. This paper develops a design-based standard error estimator for 2TS sampling. It is shown that the Taylor series approximation to the variance of the sample mean under 2TS sampling may be expressed in terms of either a deterministic variogram or a deterministic covariance function. Variance estimation then can be approached through the estimation of a variogram or a covariance function. The resulting standard error estimators are compared to some more traditional variance estimators through a simulation study. The simulation results show that estimators based on the new approach may perform better than traditional variance estimators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bartlett, R. F., 1986, Estimating the total of a continuous population: Jour. Statist. Planning and Inference, v. 13, no. 1, p. 51–66.
Cassel, C. M., Särndal, C. E., and Wretman, J. H., 1977, Foundations of inference in survey sampling: John Wiley & Sons, New York, 192 p.
Cochran, W. G., 1977, Sampling techniques (3rd. ed.): John Wiley & Sons, New York, 428 p.
Cordy, C. B., 1993, An extension of the Horvitz-Thompson theorem to point sampling from a continuous universe: Statistics and Probability Letters, v. 18, no. 5, p. 353–362.
Cressie, N., 1985, Fitting variogram models by weighted least squares: Math. Geology, v. 17, no. 5, p. 563–586.
Cressie, N., 1991, Statistics for spatial data: John Wiley & Sons, New York, 900 p.
De Gruijter, J. J., and Ter Braak, C. J. F., 1990, Model free estimation from survey samples: a reappraisal of classical sampling theory: Math. Geology, v. 22, no. 4, p. 407–415.
Fuller, W. A., 1970, Sampling with random stratum boundaries: Jour. Roy. Statis. Soc. B, v. 32, no. 2, p. 209–226.
Hansen, M. H., Madow, W. G., and Tepping, B. J., 1983, An evaluation of model-dependent and probability sampling inferences in sample surveys: Jour. Am. Statist. Soc, v. 78, no. 384, p. 776–793.
Horvitz, D. G., and Thompson, D. J., 1952, A generalisation of sampling with replacement from a finite universe: Jour. Am. Statist. Assoc., v. 47, no. 260, p. 663–685.
Iachan, R., 1982, Systematic sampling: a critical review: Intern. Statist. Review, v. 50, no. 3. pp. 293–303.
Iachan, R., 1985, Plane sampling: Statistics and Probability Letters, v. 3, no. 3, p. 151–159.
Isaaks, E. H., and Srivastava, R. M., 1988, Spatial continuity measures for probabilistic and deterministic geostatistics: Math. Geology, v. 20, no. 4, p. 313–341.
Journel, A. G., 1985, The deterministic side of geostatistics: Math. Geology, v. 17, no. 1, p. 1–15.
Koop, J. C., 1990, Systematic sampling of two-dimensional surfaces and related problems: Communications in Statistics Theory and Methods, v. 19, no. 5, p. 1701–1750.
McArthur, R. D., 1987, An evaluation of sample designs for estimating a locally concentrated pollutant: Communications in Statistics Computation and Simulation, v. 16, no. 3, p. 735–759.
Overton, W. S., and Stehman, S. V., 1993, Properties of designs for sampling continuous spatial resources from a triangular grid: Communications in Statistics Theory and Methods, v. 22, no. 9, p. 2641–2660.
Quenouille, M. H., 1949, Problems in plane sampling: Ann. Math. Statistics, v. 20, no. 3, p. 355–375.
Särndal, C. E., Swensson, B., and Wretman, J., 1992, Model assisted survey sampling: Springer-Verlag. New York, 694 p.
Smith, T. M. F., 1976, The foundations of survey sampling: a review: Jour. Roy. Statist. Soc. A, v. 139, no. 2, p. 183–204.
Wolter, K. M., 1985, Introduction to variance estimation: Springer-Verlag, New York, 427 p.
Yates, F., and Grundy, P. M., 1953, Selection without replacement from within strata with probability proportional to size: Jour. Roy. Statist. Soc. B, v. 15, no. 2, p. 253–261.
Zubrzycki, S., 1958, Remarks on stratified and systematic sampling in a plane: Colloquium Mathematicum, v. 4, p. 251–264.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cordy, C.B., Thompson, C.M. An application of the deterministic variogram to design-based variance estimation. Math Geol 27, 173–205 (1995). https://doi.org/10.1007/BF02083210
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02083210