The proposal consists of five important steps. First, the investigator develops new dependence measures (quotient correlation) and analyze the resulting test statistics. It is important to develop test statistics for tail independence based as much as possible on appropriate tail observations and less on observations from the center which may bias the analysis. Clearly, the decision on "where does appropriate" stands has to be carefully analyzed. Second, a new measure for extreme co-movement is introduced. This measure allows one to calculate probabilities of occurrences of certain extreme events in the future given the past history of the extreme movement of underlying variables. Third, the proposal includes the development of statistical estimation methods for max-stable processes. This allows one to efficiently study clustered spatial-temporal extreme observations. Fourth, procedures of how to calculate portfolio risk measures -- such as unconditional or conditional Value at Risk -- are also introduced using combined Markov process and max-stable process models. Fifth, the statistical analysis of extreme co-movements is of considerable importance in practice. For example, the notion of spillover in global financial markets and credit loss data can be thought of as one area of application.
The intellectual merit of the proposal in a first instance stems from a precise testing procedure for extremal dependence. As all definitions used depend on some limit procedures and often concern statistical testing for parameters at the edge of a specific parametric space (hence possibly testing is a non-regular estimation problem), great care has to be taken to obtain tests with sufficient power. The proposal is exactly aiming at finding a solution for this. Several procedures for tackling this problem have been published, however, up to now, no clear winning approach seems to exist. The investigator will carefully compare and contrast new approaches with existing ones using simulated as well as real data. The real data will come from areas as diverse as insurance, finance, telecommunications, climatology, seismology, medicine, etc. Beyond these methodological merits and specific applications, the proposal also has a considerable broad impact. Throughout applications in diverse fields (like above), extreme risks play an important scientific, societal as well as (possibly) political role. The dissemination of new statistical tools leading to a better understanding of the occurrence of joint extremes is of great importance. This can be very well achieved at the level of new graduate courses, publications in journals aimed at a broaden audience and in discussion with scientists from other fields.
