Abstract
Gaussian random fields are commonly used as models for spatial processes and maximum likelihood is a preferred method of choice for estimating the covariance parameters. However if the sample size n is large, evaluating the likelihood can be a numerical challenge. Covariance tapering is a way of approximating the covariance function with a taper (usually a compactly supported function) so that the computational burden is reduced. This article studies the fixed-domain asymptotic behavior of the tapered MLE for the microergodic parameter of a Matérn covariance function when the taper support is allowed to shrink as n→∞. In particular if the dimension of the underlying space is ≤3, conditions are established in which the tapered MLE is strongly consistent and also asymptotically normal. Numerical experiments are reported that gauge the quality of these approximations for finite n.
Highlights
Let X : Rd → R be a mean-zero isotropic Gaussian random field with the Matern covariance functionCov(X(x), X(y)) = σ2Kα(x − y) =σ2(α x − y 2ν−1Γ(ν) )ν Kν (α x−y ), ∀x, y ∈ Rd, (1)where ν > 0 is a known constant, α, σ are strictly positive but unknown parameters and Kν is the modified Bessel function of the second kind. . denotes the usual Euclidean norm in Rd
Σ12,nα21ν → σ2α2ν, as n → ∞, with Pα,σ probability 1 where Pα,σ is the Gaussian measure defined by the covariance function σ2Kα in (1)
The main reasons are that the proofs use the well developed theory of equivalence of Gaussian measures and that the spectral density of the Matern covariance function has a rather simple form
Summary
Let X : Rd → R be a mean-zero isotropic Gaussian random field with the Matern covariance function. Σ12,nα21ν → σ2α2ν , as n → ∞, with Pα,σ probability 1 where Pα,σ is the Gaussian measure defined by the covariance function σ2Kα in (1). The main reasons are that the proofs use the well developed theory of equivalence of Gaussian measures and that the spectral density of the Matern covariance function has a rather simple form. Zhang [19], page 259, has a discussion on the difficulties of obtaining analogous results for non-Gaussian random fields and the use of other covariance functions. The latter would be an important direction for future research. F ≍ g as w → ∞ (or n → ∞) means there exist constants 0 < c < C such that c ≤ f /g ≤ C for sufficiently large w (or n) respectively
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