Inference Problems in Extreme Value Statistics

The study of extreme-value theory has been paid much attention in recent years. Estimating probabilities of rare events is one of the primary interests. This has motivated many researchers to develop new methodologies in extreme-value statistics. One of the difficulties in applying the extreme-value theory is that the sample fraction has to be carefully chosen such that the estimation has a convergence rate as

fast as possible while its bias is negligible. This proposal consists of five topics, as follows. First, the investigator proposes a data tilting method to construct confidence intervals for extreme tail probabilities when the underlying distribution belongs to the domain of attraction for one of the extreme-value distributions. The proposed method is expected to generate more accurate confidence intervals in terms of coverage probabilities and to be more robust against the choice of the sample fraction. Second, the investigator develops new methods for estimation of dependence structures in bivariate extremes. Estimation of the dependence structures, such as the spectral measure, in bivariate extreme-value statistics is an important issue. The spectral measure, together with the two marginal limits, determines the limiting distribution of the bivariate extremes. Third, the investigator proposes smooth estimators for the first partial derivatives of the dependence function in order to construct confidence intervals for the dependence function based on the normal approximation. Fourth, the investigator studies how to construct confidence bands for the spectral measure and tail dependence functions in bivariate extreme-value statistics. Special bootstrap techniques are applied to solve the problems. This allows one to obtain asymptotically correct confidence bands without estimating the derivatives of the dependence function globally. Fifth, the investigator proposes new estimators for the Pickands dependence function of high dimensional extreme-value distributions with unknown marginal distributions.

Extreme-value statistics have found applications in many fields such as meteorology, hydrology, climatology, environmental sciences, telecommunications, insurance, and finance. The investigator develops more accurate and effective methodologies for risk analysis in both univariate and multivariate extreme-value statistics. Progress of the projects in this proposal enhances the collaboration between the investigator and researchers in these fields. The proposed activities also involve teaching graduate students to use extreme-value statistics in their future research. The new methods developed in this proposal are expected to have broader applications as well. For example, they can be used by actuaries to calculate and insure against the probability of rare but financially devastating events, or be employed by statisticians to calculate the required height of sea walls to prevent flooding. They can also be used to tell engineers how strong to build bridges or oil rigs and to model excessively high pollution levels.