Spatially varying extreme events such as storms, drought, frosts and heat waves have obvious human and financial costs. This motivates the need for probability models that sufficiently characterize the distribution of such extreme processes in order to manage the risk associated with them. A class of probability models known as max-stable models have displayed flexibility as well as theoretical justification to model extreme spatial processes. However, the calibration of such models to existing data remains a challenge. This is primarily due to the intractability of the likelihood for max-stable models. This intractability makes common likelihood based methods, such as maximum likelihood estimation and Bayesian inference, infeasible. The few existing methods for fitting max-stable models rely on composite scores of pseudo-likelihoods that are not able to capture the full dependence structure of spatially varying extremes. Given this limitation, we propose a new method of inference based on cumulative distribution functions (CDFs).
Methods and Results
Finite dimensional cumulative distribution functions of max-stable models are mathematically tractable and fully characterize extreme spatial dependence. For this reason we develop a minimum distance estimator (MDE) based on CDFs. Minimum distance estimators fit the model that minimizes a suitable distance criterion between the model CDF and empirical CDF constructed from observed data. We develop a distance criterion for CDFs in the max-stable setting and provide sufficient regularity conditions for the consistency and asymptotic normality of the MDE. We validate our methodology by conducting a simulation study. The results of the simulation study confirm the consistency and asymptotic normality of the MDE. While the MDE captures the full dependence structure of max-stable models, it is more computationally intensive than existing composite likelihood methods.
Conclusions
We develop a minimum distance estimator for max-stable models that is based on a full characterization of the spatial dependence structure. This is was not possible through previous composite likelihood based methods. We show that the resulting estimator is consistent and asymptotically normal.