Mathematical Geology

, Volume 32, Issue 1, pp 127–137

The Correlation Structure of Matheron's Classical Variogram Estimator Under Elliptically Contoured Distributions

  • Marc G. Genton
Article

Abstract

The classical variogram estimator proposed by Matheron can be written as a quadratic form of the observations. When data have an elliptically contoured distribution with constant mean, the correlation between the classical variogram estimator at two different lags is a function of the spatial design matrix, the covariance matrix, and the kurtosis. Several specific cases are studied closely. A subclass of elliptically contoured distributions with a particular family of covariance matrices is shown to possess exactly the same correlation structure for the classical variogram estimator as the multivariate independent Gaussian distribution. The consequences on variogram fitting by generalized least squares are discussed.

variogram estimation quadratic form kurtosis variogram fitting generalized least squares 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. Bentler, P. M., and Berkane, M., 1986, Greatest lower bound to the elliptical theory kurtosis parameter: Biometrika, v. 73, no. 1, p. 240–241.Google Scholar
  2. Cressie, N., and Hawkins, D. M., 1980, Robust estimation of the variogram, I: Math. Geology, v. 12, no. 2, p. 115–125.Google Scholar
  3. Cressie, N., 1993, Statistics for spatial data: John Wiley & Sons, New York, 900 p.Google Scholar
  4. Fang, K.-T., Kotz, S., and Ng, K.-W., 1989, Symmetric multivariate and related distributions: Chapman & Hall, London, 200 p.Google Scholar
  5. Fang, K.-T., and Zhang, Y.-T., 1990, Generalized Multivariate Analysis: Springer, Berlin, 220 p.Google Scholar
  6. Fang, K.-T., and Anderson, T. W., 1990, Statistical Inference in Elliptically Contoured and Related Distributions: Allerton Press, New York, 498 p.Google Scholar
  7. Genton, M. G., 1998a, Highly robust variogram estimation: Math. Geology, v. 30, no. 2, p. 213–221.Google Scholar
  8. Genton, M. G., 1998b, Variogram fitting by generalized least squares using an explicit formula for the covariance structure: Math. Geology, v. 30, no. 4, p. 323–345.Google Scholar
  9. Genton, M. G., 1998c, Spatial breakdown point of variogram estimators: Math. Geology, v. 30, no. 7, p. 853–871.Google Scholar
  10. Horn, R. A., and Johnson, C. R., 1991, Topics in matrix analysis: Cambridge University Press, Cambridge, 607 p.Google Scholar
  11. Journel, A. G., and Huijbregts, Ch. J., 1978, Mining geostatistics: Academic Press, London, 600 p.Google Scholar
  12. Li, G., 1987, Moments of a random vector and its quadratic forms: Jour. Stat. Appl. Prob., v. 2, p. 219–229.Google Scholar
  13. Li, R.-Z., Fang, K.-T., and Zhu, L.-X., 1997, Some Q-Q probability plots to test spherical and elliptical symmetry: Jour. Comp. Graph. Stat., v. 6, p. 435–450.Google Scholar
  14. Matheron, G., 1962, Traité de géostatistique appliquée, Tome I: Mémoires du Bureau de Recherches Géologiques et Miniéres, no. 14, Editions Technip, Paris, 333 p.Google Scholar
  15. Muirhead, R. J., 1982, Aspects of multivariate statistical theory: Wiley, New York, 673 p.Google Scholar
  16. Muirhead, R. J., and Waternaux, C. M., 1980, Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations: Biometrika, v. 67, p. 31–43.Google Scholar
  17. Waternaux, C. M., 1976, Asymptotic distribution of the sample roots for a nonnormal population: Biometrika, v. 63, p. 639–646.Google Scholar

Copyright information

© International Association for Mathematical Geology 2000

Authors and Affiliations

  • Marc G. Genton
    • 1
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations