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

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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.

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REFERENCES

  • 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 

  • Cressie, N., and Hawkins, D. M., 1980, Robust estimation of the variogram, I: Math. Geology, v. 12, no. 2, p. 115–125.

    Google Scholar 

  • Cressie, N., 1993, Statistics for spatial data: John Wiley & Sons, New York, 900 p.

    Google Scholar 

  • Fang, K.-T., Kotz, S., and Ng, K.-W., 1989, Symmetric multivariate and related distributions: Chapman & Hall, London, 200 p.

    Google Scholar 

  • Fang, K.-T., and Zhang, Y.-T., 1990, Generalized Multivariate Analysis: Springer, Berlin, 220 p.

    Google Scholar 

  • Fang, K.-T., and Anderson, T. W., 1990, Statistical Inference in Elliptically Contoured and Related Distributions: Allerton Press, New York, 498 p.

    Google Scholar 

  • Genton, M. G., 1998a, Highly robust variogram estimation: Math. Geology, v. 30, no. 2, p. 213–221.

    Google Scholar 

  • 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 

  • Genton, M. G., 1998c, Spatial breakdown point of variogram estimators: Math. Geology, v. 30, no. 7, p. 853–871.

    Google Scholar 

  • Horn, R. A., and Johnson, C. R., 1991, Topics in matrix analysis: Cambridge University Press, Cambridge, 607 p.

    Google Scholar 

  • Journel, A. G., and Huijbregts, Ch. J., 1978, Mining geostatistics: Academic Press, London, 600 p.

    Google Scholar 

  • Li, G., 1987, Moments of a random vector and its quadratic forms: Jour. Stat. Appl. Prob., v. 2, p. 219–229.

    Google Scholar 

  • 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 

  • 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 

  • Muirhead, R. J., 1982, Aspects of multivariate statistical theory: Wiley, New York, 673 p.

    Google Scholar 

  • 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 

  • Waternaux, C. M., 1976, Asymptotic distribution of the sample roots for a nonnormal population: Biometrika, v. 63, p. 639–646.

    Google Scholar 

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Genton, M.G. The Correlation Structure of Matheron's Classical Variogram Estimator Under Elliptically Contoured Distributions. Mathematical Geology 32, 127–137 (2000). https://doi.org/10.1023/A:1007511019496

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