Abstract
Most geostatistical applications use covariance functions that display short-range dependence, in part due to the wide variety and availability of these models in statistical packages, and in part due to spatial interpolation being the main goal of many analyses. But when the goal is spatial extrapolation or prediction based on sparsely located data, covariance functions that display long-range dependence may be more adequate. This paper constructs a new family of isotropic correlation functions whose members display long-range dependence and can also model different degrees of smoothness. This family is compared to a sub-family of the Matérn family commonly used in geostatistics, and two other recently proposed families of covariance functions with long-range dependence are discussed.
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Notes
- 1.
The symbol \(t_{\nu }(\mu ,\sigma ^2)\) denotes the t distribution with \(\nu \) degrees of freedom, location parameter \(\mu \) and scale parameter \(\sigma \).
- 2.
The fact \(K_{\theta , m}(0) = 1\) follows by continuity.
- 3.
Differentiability of \(K(\cdot )\) at zero refers to differentiability of its even extension over the real line, defined as \(K^{e}(r) := K(|r|)\), \(r \in {\mathbb R}\). Also, the phrase ‘\(Z(\cdot )\) is 0-times mean square differentiable’ is used if \(Z(\cdot )\) is mean square continuous.
- 4.
The symbol IG\((\alpha ,\beta ^2 /2)\) denotes the inverse gamma distribution with shape parameter \(\alpha \) and scale parameter \(\beta ^2 /2\).
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Acknowledgements
This work was partially supported by the U.S. National Science Foundation grant DMS–2113375.
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De Oliveira, V. (2023). A Simple Isotropic Correlation Family in \({\mathbb R}^3\) with Long-Range Dependence and Flexible Smoothness. In: Liu, Y., Hirukawa, J., Kakizawa, Y. (eds) Research Papers in Statistical Inference for Time Series and Related Models. Springer, Singapore. https://doi.org/10.1007/978-981-99-0803-5_4
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