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Spatial design matrices and associated quadratic forms: structure and properties

  • Grant Hillier
  • Federico Martellosio

The paper provides significant simplifications and extensions of results obtained by Gorsich, Genton, and Strang (J. Multivariate Anal. 80 (2002) 138) on the structure of spatial design matrices. These are the matrices implicitly defined by quadratic forms that arise naturally in modelling intrinsically stationary and isotropic spatial processes. We give concise structural formulae for these matrices, and simple generating functions for them. The generating functions provide formulae for the cumulants of the quadratic forms of interest when the process is Gaussian, second-order stationary and isotropic. We use these to study the statistical properties of the associated quadratic forms, in particular those of the classical variogram estimator, under several assumptions about the actual variogram.

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File URL: http://cemmap.ifs.org.uk/wps/cwp0416.pdf
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Paper provided by Centre for Microdata Methods and Practice, Institute for Fiscal Studies in its series CeMMAP working papers with number CWP16/04.

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Date of creation: Dec 2004
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Handle: RePEc:ifs:cemmap:16/04
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  1. Gorsich, David J. & Genton, Marc G. & Strang, Gilbert, 2002. "Eigenstructures of Spatial Design Matrices," Journal of Multivariate Analysis, Elsevier, vol. 80(1), pages 138-165, January.
  2. Ali, Mukhtar M, 1987. "Durbin-Watson and Generalized Durbin-Watson Tests for Autocorrelations and Randomness," Journal of Business & Economic Statistics, American Statistical Association, vol. 5(2), pages 195-203, April.
  3. Hillier, Grant, 2001. "THE DENSITY OF A QUADRATIC FORM IN A VECTOR UNIFORMLY DISTRIBUTED ON THE n-SPHERE," Econometric Theory, Cambridge University Press, vol. 17(01), pages 1-28, February.
  4. Hillier, G.H., 1999. "The density of a quadratic form in a vector uniformly distributed on the n-sphere," Discussion Paper Series In Economics And Econometrics 9902, Economics Division, School of Social Sciences, University of Southampton.
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