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.
Cressie, N., and Hawkins, D. M., 1980, Robust estimation of the variogram, I: Math. Geology, v. 12, no. 2, p. 115–125.
Cressie, N., 1993, Statistics for spatial data: John Wiley & Sons, New York, 900 p.
Fang, K.-T., Kotz, S., and Ng, K.-W., 1989, Symmetric multivariate and related distributions: Chapman & Hall, London, 200 p.
Fang, K.-T., and Zhang, Y.-T., 1990, Generalized Multivariate Analysis: Springer, Berlin, 220 p.
Fang, K.-T., and Anderson, T. W., 1990, Statistical Inference in Elliptically Contoured and Related Distributions: Allerton Press, New York, 498 p.
Genton, M. G., 1998a, Highly robust variogram estimation: Math. Geology, v. 30, no. 2, p. 213–221.
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.
Genton, M. G., 1998c, Spatial breakdown point of variogram estimators: Math. Geology, v. 30, no. 7, p. 853–871.
Horn, R. A., and Johnson, C. R., 1991, Topics in matrix analysis: Cambridge University Press, Cambridge, 607 p.
Journel, A. G., and Huijbregts, Ch. J., 1978, Mining geostatistics: Academic Press, London, 600 p.
Li, G., 1987, Moments of a random vector and its quadratic forms: Jour. Stat. Appl. Prob., v. 2, p. 219–229.
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.
Matheron, G., 1962, Traité de géostatistique appliquée, Tome I: Mémoires du Bureau de Recherches Géologiques et Miniéres, no. 14, Editions Technip, Paris, 333 p.
Muirhead, R. J., 1982, Aspects of multivariate statistical theory: Wiley, New York, 673 p.
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.
Waternaux, C. M., 1976, Asymptotic distribution of the sample roots for a nonnormal population: Biometrika, v. 63, p. 639–646.
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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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DOI: https://doi.org/10.1023/A:1007511019496