Title: Infinitesimal perturbation analysis for risk measures based on the Smith max-stable random field,
Abstract: 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 very 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.