Abstract
This article considers estimating the smoothness parameter of a class of nonstationary Gaussian random fields on Rd using irregularly spaced data observed along a curve. The set of covariance functions includes a nonstationary version of the Matérn covariance function as well as isotropic Matérn covariance function. Smoothness estimators are constructed via higher-order quadratic variations. Under mild conditions, these estimators are shown to be strongly consistent and convergence rate upper bounds are established with respect to fixed-domain asymptotics. Simulations indicate that the proposed estimators perform well for moderate sample sizes.
Highlights
Let X(t), t ∈ Rd, be a Gaussian random field with covariance function K(x, y) = Cov{X(x), X(y)}
Motivated by regression and spatial modeling, Paciorek [22], Paciorek and Schervish [23] propose a nonstationary version of the Matern covariance function given by KP (x, y) σ2 2ν−1Γ(ν
Loh [20] discussed second-order quadratic variations from a sample of Gaussian random field observations taken along a smooth curve in R2
Summary
Loh [20] discussed second-order quadratic variations from a sample of Gaussian random field observations taken along a smooth curve in R2. This article is motivated by [20] where the idea of using higher-order quadratic variations Vθ, , θ ∈ {1, 2}, ∈ Z+, for constructing smoothness estimators is proposed for irregularly spaced data. The results in this article complement those in [21] with regard to the estimation of the smoothness parameter of an isotropic Gaussian random field with a Matern covariance function. [8, 10, 16] all assume that ν ∈ (0, 1) Another example of spaced data on an interval is [15] where higher-order quadratic variations are used to construct a consistent estimate for ν given that ν ∈ (D, D + 1) for some known integer D. Appendices A to F contain the proofs of all the results in this article
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