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The role of the range parameter for estimation and prediction in geostatistics

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  • C. G. Kaufman
  • B. A. Shaby

Abstract

Two canonical problems in geostatistics are estimating the parameters in a specified family of stochastic process models and predicting the process at new locations. We show that asymptotic results for a Gaussian process over a fixed domain with Matérn covariance function, previously proven only in the case of a fixed range parameter, can be extended to the case of jointly estimating the range and the variance of the process. Moreover, we show that intuition and approximations derived from asymptotics using a fixed range parameter can be problematic when applied to finite samples, even for large sample sizes. In contrast, we show via simulation that performance is improved and asymptotic approximations are applicable for smaller sample sizes when the parameters are jointly estimated. These effects are particularly apparent when the process is mean square differentiable or the effective range of spatial correlation is small. Copyright 2013, Oxford University Press.

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  • C. G. Kaufman & B. A. Shaby, 2013. "The role of the range parameter for estimation and prediction in geostatistics," Biometrika, Biometrika Trust, vol. 100(2), pages 473-484.
    • Handle: RePEc:oup:biomet:v:100:y:2013:i:2:p:473-484
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    File URL: http://hdl.handle.net/10.1093/biomet/ass079
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    Cited by:

    1. Girard, Didier A., 2016. "Asymptotic near-efficiency of the “Gibbs-energy and empirical-variance” estimating functions for fitting Matérn models — I: Densely sampled processes," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 191-197.
    2. Saeed Moradi & Ali Nejat, 2020. "RecovUS: An Agent-Based Model of Post-Disaster Household Recovery," Journal of Artificial Societies and Social Simulation, Journal of Artificial Societies and Social Simulation, vol. 23(4), pages 1-13.
    3. Lim, Chae Young & Chen, Chien-Hung & Wu, Wei-Ying, 2017. "Numerical instability of calculating inverse of spatial covariance matrices," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 182-188.
    4. Bachoc, François & Lagnoux, Agnès & Nguyen, Thi Mong Ngoc, 2017. "Cross-validation estimation of covariance parameters under fixed-domain asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 42-67.
    5. Hong, Yiping & Zhou, Zaiying & Yang, Ying, 2020. "Hypothesis testing for the smoothness parameter of Matérn covariance model on a regular grid," Journal of Multivariate Analysis, Elsevier, vol. 177(C).
    6. Wenpin Tang & Lu Zhang & Sudipto Banerjee, 2021. "On identifiability and consistency of the nugget in Gaussian spatial process models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(5), pages 1044-1070, November.
    7. Bachoc, François & Bevilacqua, Moreno & Velandia, Daira, 2019. "Composite likelihood estimation for a Gaussian process under fixed domain asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    8. Girard, Didier A., 2020. "Asymptotic near-efficiency of the “Gibbs-energy (GE) and empirical-variance” estimating functions for fitting Matérn models - II: Accounting for measurement errors via “conditional GE mean”," Statistics & Probability Letters, Elsevier, vol. 162(C).
    9. Sierra Pugh & Matthew J. Heaton & Jeff Svedin & Neil Hansen, 2019. "Spatiotemporal Lagged Models for Variable Rate Irrigation in Agriculture," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 24(4), pages 634-650, December.
    10. Bevilacqua, Moreno & Caamaño-Carrillo, Christian & Porcu, Emilio, 2022. "Unifying compactly supported and Matérn covariance functions in spatial statistics," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    11. Victor De Oliveira & Zifei Han, 2022. "On Information About Covariance Parameters in Gaussian Matérn Random Fields," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 27(4), pages 690-712, December.
    12. Martin Tingley & Benjamin Shaby, 2015. "Comments on: Comparing and selecting spatial predictors using local criteria," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(1), pages 47-53, March.

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