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Stochastic derivative estimation for max-stable random fields

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  • Koch, Erwan
  • Robert, Christian Y.

Abstract

We consider expected performances based on max-stable random fields and we are interested in their derivatives with respect to the spatial dependence parameters of those fields. Max-stable fields, such as the Brown–Resnick and Smith fields, are very popular in spatial extremes. We focus on the two most popular unbiased stochastic derivative estimation approaches: the likelihood ratio method (LRM) and the infinitesimal perturbation analysis (IPA). LRM requires the multivariate density of the max-stable field to be explicit, and IPA necessitates the computation of the derivative with respect to the parameters for each simulated value. We propose convenient and tractable conditions ensuring the validity of LRM and IPA in the cases of the Brown–Resnick and Smith field, respectively. Obtaining such conditions is intricate owing to the very structure of max-stable fields. Then we focus on risk and dependence measures, which constitute one of the several frameworks where our theoretical results can be useful. We perform a simulation study which shows that both LRM and IPA perform well in various configurations, and provide a real case study that is valuable for the insurance industry.

Suggested Citation

  • Koch, Erwan & Robert, Christian Y., 2022. "Stochastic derivative estimation for max-stable random fields," European Journal of Operational Research, Elsevier, vol. 302(2), pages 575-588.
    • Handle: RePEc:eee:ejores:v:302:y:2022:i:2:p:575-588
    DOI: 10.1016/j.ejor.2021.12.026
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    References listed on IDEAS

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    1. Paul Glasserman & Zongjian Liu, 2010. "Sensitivity Estimates from Characteristic Functions," Operations Research, INFORMS, vol. 58(6), pages 1611-1623, December.
    2. Kerry Emanuel, 2005. "Increasing destructiveness of tropical cyclones over the past 30 years," Nature, Nature, vol. 436(7051), pages 686-688, August.
    3. Nan Chen & Yanchu Liu, 2014. "American Option Sensitivities Estimation via a Generalized Infinitesimal Perturbation Analysis Approach," Operations Research, INFORMS, vol. 62(3), pages 616-632, June.
    4. Clément Dombry & Sebastian Engelke & Marco Oesting, 2016. "Exact simulation of max-stable processes," Biometrika, Biometrika Trust, vol. 103(2), pages 303-317.
    5. Opitz, T., 2013. "Extremal t processes: Elliptical domain of attraction and a spectral representation," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 409-413.
    6. Martin Schlather, 2003. "A dependence measure for multivariate and spatial extreme values: Properties and inference," Biometrika, Biometrika Trust, vol. 90(1), pages 139-156, March.
    7. Cristiano Varin & Paolo Vidoni, 2005. "A note on composite likelihood inference and model selection," Biometrika, Biometrika Trust, vol. 92(3), pages 519-528, September.
    8. Mark Broadie & Paul Glasserman, 1996. "Estimating Security Price Derivatives Using Simulation," Management Science, INFORMS, vol. 42(2), pages 269-285, February.
    9. Marc G. Genton & Yanyuan Ma & Huiyan Sang, 2011. "On the likelihood function of Gaussian max-stable processes," Biometrika, Biometrika Trust, vol. 98(2), pages 481-488.
    10. Erwan Koch, 2018. "Extremal dependence and spatial risk measures for insured losses due to extreme winds," Papers 1804.05694, arXiv.org, revised Dec 2019.
    11. C. Dombry & F. Éyi-Minko & M. Ribatet, 2013. "Conditional simulation of max-stable processes," Biometrika, Biometrika Trust, vol. 100(1), pages 111-124.
    12. Heidergott, Bernd & Vazquez-Abad, Felisa J. & Volk-Makarewicz, Warren, 2008. "Sensitivity estimation for Gaussian systems," European Journal of Operational Research, Elsevier, vol. 187(1), pages 193-207, May.
    13. B. Heidergott & F. J. Vázquez-Abad, 2008. "Measure-Valued Differentiation for Markov Chains," Journal of Optimization Theory and Applications, Springer, vol. 136(2), pages 187-209, February.
    14. Padoan, S. A. & Ribatet, M. & Sisson, S. A., 2010. "Likelihood-Based Inference for Max-Stable Processes," Journal of the American Statistical Association, American Statistical Association, vol. 105(489), pages 263-277.
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