Infinitesimal perturbation analysis for risk measures based on the Smith max-stable random field
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.