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Towards reconciling two asymptotic frameworks in spatial statistics

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  • Hao Zhang
  • Dale L. Zimmerman

Abstract

Two asymptotic frameworks, increasing domain asymptotics and infill asymptotics, have been advanced for obtaining limiting distributions of maximum likelihood estimators of covariance parameters in Gaussian spatial models with or without a nugget effect. These limiting distributions are known to be different in some cases. It is therefore of interest to know, for a given finite sample, which framework is more appropriate. We consider the possibility of making this choice on the basis of how well the limiting distributions obtained under each framework approximate their finite-sample counterparts. We investigate the quality of these approximations both theoretically and empirically, showing that, for certain consistently estimable parameters of exponential covariograms, approximations corresponding to the two frameworks perform about equally well. For those parameters that cannot be estimated consistently, however, the infill asymptotic approximation is preferable. Copyright 2005, Oxford University Press.

Suggested Citation

  • Hao Zhang & Dale L. Zimmerman, 2005. "Towards reconciling two asymptotic frameworks in spatial statistics," Biometrika, Biometrika Trust, vol. 92(4), pages 921-936, December.
    • Handle: RePEc:oup:biomet:v:92:y:2005:i:4:p:921-936
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    File URL: http://hdl.handle.net/10.1093/biomet/92.4.921
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    Citations

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    Cited by:

    1. José León & Carenne Ludeña, 2015. "Difference based estimators and infill statistics," Statistical Inference for Stochastic Processes, Springer, vol. 18(1), pages 1-31, April.
    2. Christopher J. Geoga & Mihai Anitescu & Michael L. Stein, 2021. "Flexible nonstationary spatiotemporal modeling of high‐frequency monitoring data," Environmetrics, John Wiley & Sons, Ltd., vol. 32(5), August.
    3. Maroussa Zagoraiou & Alessandro Baldi Antognini, 2009. "Optimal designs for parameter estimation of the Ornstein–Uhlenbeck process," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(5), pages 583-600, September.
    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. Bachoc, François, 2013. "Cross Validation and Maximum Likelihood estimations of hyper-parameters of Gaussian processes with model misspecification," Computational Statistics & Data Analysis, Elsevier, vol. 66(C), pages 55-69.
    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. Kiselák, Jozef & Stehlík, Milan, 2008. "Equidistant and D-optimal designs for parameters of Ornstein-Uhlenbeck process," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1388-1396, September.
    8. 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).
    9. Alain Pirotte & Jesús Mur, 2017. "Neglected dynamics and spatial dependence on panel data: consequences for convergence of the usual static model estimators," Spatial Economic Analysis, Taylor & Francis Journals, vol. 12(2-3), pages 202-229, July.
    10. Tae Kim & Jeong Park & Gyu Song, 2010. "An asymptotic theory for the nugget estimator in spatial models," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(2), pages 181-195.
    11. Luis Sánchez & Víctor Leiva & Manuel Galea & Helton Saulo, 2020. "Birnbaum-Saunders Quantile Regression Models with Application to Spatial Data," Mathematics, MDPI, vol. 8(6), pages 1-17, June.
    12. 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).
    13. Boukouvalas, A. & Cornford, D. & Stehlík, M., 2014. "Optimal design for correlated processes with input-dependent noise," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 1088-1102.
    14. Haozhe Zhang & Yehua Li, 2020. "Unified Principal Component Analysis for Sparse and Dense Functional Data under Spatial Dependency," Papers 2006.13489, arXiv.org, revised Jun 2021.
    15. Lim, Chae Young & Stein, Michael, 2008. "Properties of spatial cross-periodograms using fixed-domain asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 99(9), pages 1962-1984, October.
    16. 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).
    17. Wu, Wei-Ying & Lim, Chae Young & Xiao, Yimin, 2013. "Tail estimation of the spectral density for a stationary Gaussian random field," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 74-91.
    18. Arthur P. Guillaumin & Adam M. Sykulski & Sofia C. Olhede & Frederik J. Simons, 2022. "The Debiased Spatial Whittle likelihood," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(4), pages 1526-1557, September.
    19. Ganggang Xu & Marc G. Genton, 2017. "Tukey -and- Random Fields," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1236-1249, July.
    20. Werner Müller & Milan Stehlík, 2009. "Issues in the optimal design of computer simulation experiments," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(2), pages 163-177, March.

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