Samuel Mueller School of Mathematics and Statistics University of Sydney Location: Carslaw 173 Time: 2pm Friday, April 24, 2009 Title: Smooth extreme value estimation for densities with log-concave tails Abstract: It is shown that (1) both parametric distribution functions appearing in extreme value theory -- the generalized extreme value distribution and the generalized Pareto distribution -- have log-concave densities if the extreme value index is in [-1,0] and (2) that all distribution functions with log-concave density belong to the max--domain of attraction of the generalized extreme value distribution. Many extreme value tail-index estimators, such as (generalized) Pickands, are functions of the upper order statistics. Therefore, one can replace these order statistics by their corresponding smoothed quantiles from the distribution function that is based on the estimated log-concave density. This leads to smooth quantile and tail index estimators. These new estimators are particularly useful in small samples. Acting as a smoother of the empirical distribution function, the log-concave distribution function estimator reduces estimation variability to a much greater extent than it introduces bias. As a consequence, Monte Carlo simulations demonstrate that the smoothed version of the estimators are superior to their non-smoothed counterparts, in terms of mean squared error. The R package smoothtail is presented, which provides functions to calculate the smoothed estimators. References  Mueller, S., Rufibach, K. (2007). Software smoothtail: Smooth estimation of GPD shape parameter. R package version 1.1.2.  Mueller, S., Rufibach, K. (2008). On the Max-Domain of Attraction of Distributions with Log-Concave Densities. Statistics & Probability Letters78:1440-1444.  Mueller, S., Rufibach, K. (2009 to appear). Smooth Tail Index Estimators. Journal of Statistical Computation and Simulation, 13 pages.