Statistical inference on extreme values.

The fundamental question in Non-parametric Statistics is the possibility of consistent estimation of a quantity of interest. Not always consistent estimation in a non-parametric class of distributions is possible. The next key question is the accuracy of estimation. These questions can be addressed by establishing lower bounds and evaluating the rate of approximation. The proposed research will concentrate on establishing lower bounds to the accuracy of estimation of the distribution function of the sample maximum in a general non-parametric class of distributions, establishing lower bounds to the accuracy of estimation of the tail index of a heavy-tailed distribution and extreme quantiles, deriving estimates of the accuracy of compound Poisson approximation to the distribution of the empirical point processes of exceedances. Note that the problem of accurate estimation of extreme quantiles has important applications in Finance and Insurance where an extreme quantile is used as a measure of risk known as Value-at-Risk. Support is sought for the principal investigator's visits to the Department of Statistics, University of Georgia, USA, College of Engineering, University of Florida, USA, the Department of Mathematics & Statistics, University of Melbourne, Australia, and a visit of a member of staff of the College of Engineering, University of Florida, to the applicant's host university. The aim is to develop collaboration in the areas of Non-parametric Statistics and Financial Mathematics with Prof. S.Uryasev, Dr K.Borovkov and Dr A.Xia.