A Note of the Asymptotic Distribution of the Sample Semivarogram

Author: Hsiao-Chuan Lu(Department of Applied Mathematics, Chinese Culture University), Shao-Tung Chang(Department of Mathematics, National Taiwan Normal University)

Vol.&No.：Vol. 45, No. 1&2
Date：October 2000
Pages：11-19
DOI：10.6301/JNTNU.2000.45.02

Abstract：

The semivariogram is widely used in geostatistical data analysis. Most often, it is estimated by the sample semivariogram. Knowledge about the sampling distribution of the sample semivariogram is necessary in many statistical inferences, such as assessing the goodness of fit of any proposed semivariogram model, and testing for directional symmetry properties. Some theoretical works have been done on the marginal distribution of the sample semivariogram for a Gaussian m-dependent process, but it is not of immediate practical use due to the restrictive conditions and the presence of spatial inter-correlations. Although the joint distrbution of the sample semivariogram has been discussed through simulation studies, theoretical support is needed. In this paper, we prove the joint asymptotic normality of the sample semivariogram under a mixing condition which is less restrictive than m-dependence and is satisfied by most geostatistical models. The distributional result is established for random fields without Gaussianity assumption and then specialized to Gaussian random fields.

Keywords：Geostatistics, Semivariogram, Sample semivariogram, Sampling distribution

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