Using robust and Bayesian methods to assess asymptotic independence in extreme values

Quantifying dependence is an important theme in statistical methods for multivariate extremes and extremes of time series. In a limiting sense, two situations are possible: the extremes are asymptotically dependent or asymptotically independent (Sibuya, 1960). Ad hoc tests and models for asymptotic independence have been proposed. In general, we cannot be sure that the data are not contaminated by observations not following our model, especially in complicated extreme value problems. It is important to have tests and models with desirable robustness properties.Under the Coles and Pauli (2002) model of multivariate extremes, we develop an influence measure to identify points that are influential in the asymptotic dependence estimate. We then achieve robust Bayesian estimation by discarding influential points. Numerical results show that our estimates do give the required protection if the data contain outliers, and do not introduce noticeable bias if there is no contamination.An extension is to apply our techniques in multivariate extremes to pairs of successive values in a time series. We first propose a general &egr-contamination class, extending the Sisson and Coles (2003) model, to include the possibility of contamination occurring in clusters. We then provide robust Bayesian estimation of the extremal index, which measures the tendency of a Markov process to cluster at extreme levels. The amount of robustness can be assessed through the use of proposed influence measures.In summary, we have achieved robust tests and Bayesian estimators of asymptotic independence not only in multivariate extremes but also in the extremes of Markov chains. Simple diagnostic tools for evaluating the robustness of Bayesian estimators have also been developed.We develop a robust bootstrapped test of asymptotic independence. We use a function measuring the deviation of an observation from the assumed model to shrink the contamination and then use bootstrapped p-values. The bootstrapped p-values are used to resample the test statistic from the empirical distribution rather than from the estimated population distribution satisfying the null constraints. Simulations and applications show the good performance of the new test.Key words. asymptotic independence, Bayesian robustness, contamination, extremal index, extreme value theory, influence measure, Markov Chain Monte Carlo, mixture model, robust estimation, robust test.