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Efficient inference for spatial extreme value processes associated to log-Gaussian random functions

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  • Jennifer L. Wadsworth
  • Jonathan A. Tawn

Abstract

Max-stable processes arise as the only possible nontrivial limits for maxima of affinely normalized identically distributed stochastic processes, and thus form an important class of models for the extreme values of spatial processes. Until recently, inference for max-stable processes has been restricted to the use of pairwise composite likelihoods, due to intractability of higher-dimensional distributions. In this work we consider random fields that are in the domain of attraction of a widely used class of max-stable processes, namely those constructed via manipulation of log-Gaussian random functions. For this class, we exploit limiting d-dimensional multivariate Poisson process intensities of the underlying process for inference on all d-vectors exceeding a high marginal threshold in at least one component, employing a censoring scheme to incorporate information below the marginal threshold. We also consider the d-dimensional distributions for the equivalent max-stable process, and perform full likelihood inference by exploiting the methods of Stephenson & Tawn (2005), where information on the occurrence times of extreme events is shown to dramatically simplify the likelihood. The Stephenson–Tawn likelihood is in fact simply a special case of the censored Poisson process likelihood. We assess the improvements in inference from both methods over pairwise likelihood methodology by simulation.

Suggested Citation

  • Jennifer L. Wadsworth & Jonathan A. Tawn, 2014. "Efficient inference for spatial extreme value processes associated to log-Gaussian random functions," Biometrika, Biometrika Trust, vol. 101(1), pages 1-15.
  • Handle: RePEc:oup:biomet:v:101:y:2014:i:1:p:1-15.
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    File URL: http://hdl.handle.net/10.1093/biomet/ast042
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    Cited by:

    1. R de Fondeville & A C Davison, 2018. "High-dimensional peaks-over-threshold inference," Biometrika, Biometrika Trust, vol. 105(3), pages 575-592.
    2. Rishikesh Yadav & Raphaël Huser & Thomas Opitz, 2021. "Spatial hierarchical modeling of threshold exceedances using rate mixtures," Environmetrics, John Wiley & Sons, Ltd., vol. 32(3), May.
    3. Moosup Kim & Sangyeol Lee, 2022. "Maximum composite likelihood estimation for spatial extremes models of Brown–Resnick type with application to precipitation data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 1023-1059, September.
    4. Alexis Bienvenüe & Christian Y. Robert, 2017. "Likelihood Inference for Multivariate Extreme Value Distributions Whose Spectral Vectors have known Conditional Distributions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(1), pages 130-149, March.
    5. John H. J. Einmahl & Anna Kiriliouk & Andrea Krajina & Johan Segers, 2016. "An M-estimator of spatial tail dependence," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 275-298, January.
    6. Samuel A. Morris & Brian J. Reich & Emeric Thibaud, 2019. "Exploration and Inference in Spatial Extremes Using Empirical Basis Functions," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 24(4), pages 555-572, December.
    7. Raphaël de Fondeville & Anthony C. Davison, 2022. "Functional peaks‐over‐threshold analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(4), pages 1392-1422, September.
    8. Rootzen, Holger & Segers, Johan & Wadsworth, Jenny, 2016. "Multivariate peaks over thresholds models," LIDAM Discussion Papers ISBA 2016018, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    9. Kiriliouk, Anna, 2020. "Hypothesis testing for tail dependence parameters on the boundary of the parameter space," Econometrics and Statistics, Elsevier, vol. 16(C), pages 121-135.
    10. Hentschel, Manuel & Engelke, Sebastian & Segers, Johan, 2022. "Statistical Inference for Hüsler–Reiss Graphical Models Through Matrix Completions," LIDAM Discussion Papers ISBA 2022032, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    11. Z. I. Botev, 2017. "The normal law under linear restrictions: simulation and estimation via minimax tilting," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(1), pages 125-148, January.
    12. Einmahl, John & Kiriliouk, A. & Segers, J.J.J., 2016. "A Continuous Updating Weighted Least Squares Estimator of Tail Dependence in High Dimensions," Other publications TiSEM a3e7350b-4773-4bd8-9c3c-6, Tilburg University, School of Economics and Management.
    13. Wang, Yixin & So, Mike K.P., 2016. "A Bayesian hierarchical model for spatial extremes with multiple durations," Computational Statistics & Data Analysis, Elsevier, vol. 95(C), pages 39-56.
    14. Samuel A. Morris & Brian J. Reich & Emeric Thibaud & Daniel Cooley, 2017. "A space-time skew-t model for threshold exceedances," Biometrics, The International Biometric Society, vol. 73(3), pages 749-758, September.
    15. Lee, Xing Ju & Hainy, Markus & McKeone, James P. & Drovandi, Christopher C. & Pettitt, Anthony N., 2018. "ABC model selection for spatial extremes models applied to South Australian maximum temperature data," Computational Statistics & Data Analysis, Elsevier, vol. 128(C), pages 128-144.
    16. Ho, Zhen Wai Olivier & Dombry, Clément, 2019. "Simple models for multivariate regular variation and the Hüsler–Reiß Pareto distribution," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 525-550.
    17. Jordan Richards & Jennifer L. Wadsworth, 2021. "Spatial deformation for nonstationary extremal dependence," Environmetrics, John Wiley & Sons, Ltd., vol. 32(5), August.

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