On the consistent separation of scale and variance for Gaussian random fields

By Ethan B. Anderes. 2009

77 citations recorded by the source corpus.

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 Matern autocovariance in dimension d > 4. 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 Matern Gaussian random fields with different parameters when d > 4. The case d = 4 is still open.

Subfields: Environmental Engineering, Management, Monitoring, Policy and Law, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/85b3429b31da67bbb17faead60f75b44199bec4a

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In Environmental Engineering

In Management, Monitoring, Policy and Law

In Statistical and Nonlinear Physics

Citations in the corpus

On fixed-domain asymptotics and covariance tapering in Gaussian random field models (2011, background): “In contrast for d ≥ 5, Anderes [1] recently proved that the Gaussian measures defined by σ(2)Kα and σ 2 1Kα1 are orthogonal if (α1, σ1) 6= (α, σ) and hence α and σ can be consistently estimated under fixed-domain asymptotics.”
Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics (2016, methodology): “In contrast for d ≥ 5, Anderes (2010) proved the orthogonality of two Gaussian measures with different Matérn covariance functions and hence, in this case, all the parameters can be consistently estimated under fixed-domain asymptotics. The case d = 4 is still open. Asymptotic results for ML estimat”
Constructing Priors that Penalize the Complexity of Gaussian Random Fields (2015, background): “It is unknown whether the results hold for d = 4 since it is an open question whether the parameters can be consistently estimated for that case (Anderes 2010).”