Open Access
April 2010 On the consistent separation of scale and variance for Gaussian random fields
Ethan Anderes
Ann. Statist. 38(2): 870-893 (April 2010). DOI: 10.1214/09-AOS725


We present fixed domain asymptotic results that establish consistent estimates of the variance and scale parameters for a Gaussian random field with a geometric anisotropic Matérn autocovariance in dimension d>4. When d<4 this is impossible due to the mutual absolute continuity of Matérn Gaussian random fields with different scale and variance (see Zhang [J. Amer. Statist. Assoc. 99 (2004) 250–261]). Informally, when d>4, we show that one can estimate the coefficient on the principle irregular term accurately enough to get a consistent estimate of the coefficient on the second irregular term. These two coefficients can then be used to separate the scale and variance. We extend our results to the general problem of estimating a variance and geometric anisotropy for more general autocovariance functions. Our results illustrate the interaction between the accuracy of estimation, the smoothness of the random field, the dimension of the observation space and the number of increments used for estimation. As a corollary, our results establish the orthogonality of Matérn Gaussian random fields with different parameters when d>4. The case d=4 is still open.


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Ethan Anderes. "On the consistent separation of scale and variance for Gaussian random fields." Ann. Statist. 38 (2) 870 - 893, April 2010.


Published: April 2010
First available in Project Euclid: 19 February 2010

zbMATH: 1204.60041
MathSciNet: MR2604700
Digital Object Identifier: 10.1214/09-AOS725

Primary: 60G60
Secondary: 62M30 , 62M40

Keywords: Covariance estimation , Gaussian random fields , infill asymptotics , Matérn class , principle irregular term , quadratic variations

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 2 • April 2010
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