When using risk or dependence measures based on a given underlying model, it is essential to be able to quantify the sensitivity or robustness of these measures with respect to the model parameters . In this paper, we consider an underlying model which is popular in spatial extremes , the Smith max-stable random field. We study the sensitivity properties of risk or dependence measures based on the values of this field at a finite number of locations. Max-stable fields play a key role, e.g., in the modelling of natural disasters. As their multivariate density is generally not available for more than three locations, the likelihood ratio method cannot be used to estimate the derivatives of the risk measures with respect to the model parameters . Thus, we focus on a pathwise method, the infinitesimal perturbation analysis (IPA). We provide a convenient and tractable sufficient condition for performing IPA , which is intricate to obtain because of the very structure of max-stable fields involving pointwise maxima over an infinite number of random functions. IPA enables the consistent estimation of the considered measures' derivatives with respect to the parameters characterizing the spatial dependence. We carry out a simulation study which shows that the approach performs well in various configurations  .
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 .
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When using risk or dependence measures based on a given underlying model, it is essential to be able to quantify the sensitivity or robustness of these measures
We consider expected performances based on max-stable random fields and we are interested in their derivatives
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model parameters . In this paper, we consider an underlying model which is
spatial dependence parameters of those fields. Max-stable fields, such as the Brown--Resnick and Smith fields, are very
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, the Smith
. 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
R
random field. We study the sensitivity properties of risk or dependence measures based on the values of this field at a finite number of locations. Max-stable fields play a key role, e.g., in the modelling of natural disasters. As their multivariate density is generally not available for more than three locations, the likelihood ratio method cannot be used to estimate the derivatives of the risk measures
field to be explicit, and IPA necessitates the computation of the derivative
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model parameters . Thus, we focus on a pathwise method, the infinitesimal perturbation analysis (IPA). We provide a
parameters for each simulated value. We propose
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sufficient condition for performing IPA , which is intricate to obtain because of
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
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involving pointwise maxima over an infinite number of random functions. IPA enables the consistent estimation of the considered measures' derivatives with respect to the parameters characterizing the spatial dependence. We carry out
. Then we focus on risk and dependence measures, which constitute one of the several frameworks where our theoretical results can be useful. We perform
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the approach performs
both LRM and IPA perform
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, and provide a real case study that is valuable for the insurance industry