Risk quantification and simulation metamodeling for extremal data

Abstract

Risk quantication in extreme events has been a crucial but challenging issue. Extreme value theory has been developed to provide sucient theoretical instruments in that regard. Even though we can quantify extremal risks based on extreme value theory, the estimation requires considerable sample generation with a huge computational cost in general. Furthermore, it is impossible to generate samples at all values of parameters of interest. Thus, kriging can be adopted to predict extreme quantiles, where kriging is a common prediction method that builds a metamodel on the assumption that nearby outputs are correlated. The main purpose of this dissertation is to investigate the estimation and prediction methods of extreme quantile in spite of the eective data sparsity.Firstly, a correct specication of the extremal dependence structure is difficult due to the limited size of eective data. We study the problem of estimating quantiles for bivariate extreme value distributions, considering that an estimated Pickands dependence function may deviate from the truth within some xed distance. Secondly, we propose a new metamodeling technique, extremal kriging, to predict extreme quantiles. The extreme quantile is a crucial risk measure in high-tech manufacturing or in environmental sciences. However, it is dicult to obtain a reliable response surface with only one value of estimated extreme quantile at each design point. To resolve such diculties, we construct two separate response surfaces of intermediate order statistics and extreme value indices. Then, we compute our designed response surface based on them. Thirdly, we extend the second topic to kriging method for multivariate extreme quantile prediction. Similar to extremal kriging, we propose a prediction method that performs kriging separately on the extremal dependency, intermediate order statistics and extreme value indices. This method is established on the studies on extremal dependence and kriging in the previous two topics.