Hillier, Grant and Martellosio, Federico (2006): Spatial design matrices and associated quadratic forms: structure and properties. Published in: Journal of Multivariate Analysis , Vol. 97, (2006): pp. 118.

PDF
MPRA_paper_15807.pdf Download (275kB)  Preview 
Abstract
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, secondorder 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.
Item Type:  MPRA Paper 

Original Title:  Spatial design matrices and associated quadratic forms: structure and properties 
Language:  English 
Keywords:  Cumulant; Intrinsically stationary process; Kronecker product; Quadratic form; Spatial design matrix; Variogram 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C10  General C  Mathematical and Quantitative Methods > C3  Multiple or Simultaneous Equation Models; Multiple Variables > C31  CrossSectional Models; Spatial Models; Treatment Effect Models; Quantile Regressions; Social Interaction Models C  Mathematical and Quantitative Methods > C0  General > C01  Econometrics 
Item ID:  15807 
Depositing User:  Grant Hillier 
Date Deposited:  19. Jun 2009 05:48 
Last Modified:  24. Feb 2013 15:26 
References:  [1] M.M. Ali, Durbin–Watson generalized Durbin–Watson tests for autocorrelations and randomness, J. Bus. Econom. Statist. 5 (1987) 195–203. [2] T.W. Anderson, The Statistical Analysis of Time Series, Wiley, New York, 1971. [3] M.J. Beeson, Triangles with vertices on lattice points, Amer. Math. Mon. 99 (1992) 243–252. [4] J. Besag, Spatial interaction and the statistical analysis of lattice systems, J. Roy. Statist. Soc. Ser. B 36 (1974) 192–236. [5] N. Cressie, Fitting varogram models by weighted least squares, Math. Geol. 17 (1985) 563–586. [6] N. Cressie, Statistics for Spatial Data, Wiley, New York, 1993. [7] D.M. Cvetkovi´c, M. Doob, H. Sachs, Spectra of Graphs, Academic Press, New York, 1980. [8] J. Durbin, G.S. Watson, Testing for serial correlation in least squares regression II, Biometrika 38 (1951) 159–178. [9] M.G. Genton, Variogram fitting by generalized least squares using an explicit formula for the covariance structure, Math. Geol. 30 (1998) 323–345. [10] D.J. Gorsich, M.G. Genton, G. Strang, Eigenstructures of spatial design matrices, J. Multivariate Anal. 80 (2002) 138–165.[11] G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, fifth ed., Oxford University Press, Oxford, 1979. [12] R.C. Henshaw, Testing singleequation least squares regression models for autocorrelated disturbances, Econometrica 34 (1996) 646–660. [13] G.H. Hillier, The density of a quadratic form in a vector uniformly distributed on the nsphere, Econometric Theory 17 (2001) 1–28. [14] A.T. James, Distributions of matrix variates and latent roots derived from normal samples, Ann. Math. Statist. 35 (1964) 475–501. [15] M.G. Kendall, A. Stuart, The Advanced Theory of Statistics, vol. 1, Distribution Theory, Griffin and Co., London, 1969. [16] M. Marcus, H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Dover, New York, 1969. [17] B. Mohar, Some applications of Laplace eigenvalues of graphs, in: G. Hahn, G. Sabidussi (Eds.), Graph Symmetry: Algebraic Methods and Applications, vol. 497 of NATO ASI Series C, Kluwer, Dordrecht, 1997, pp. 227–275. [18] P.A.P. Moran, Notes on continuous stochastic phenomena, Biometrika 37 (1950) 17–23. [19] J. von Neumann, R.H. Kent, H.R. Bellinson, B.I. Hart, The mean square successive differences, Ann. Math. Statist. 12 (1941) 153–162. [20] H.S. Wilf, generatingfunctionology, second ed., Academic Press Inc., New York, 1994. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/15807 