Smooth tail index estimation

Details Smooth tail index estimation Smooth tail index estimation

2006-12-06

arxiv.org/abs/math/0612140v2

Journal of Statistical Computation and Simulation 2009, Vol. 79, No. 9, 1155-1167

Mathematics

Statistics Theory

17 pages, 5 figures. Slightly changed Pickand's estimator, added some more introduction and discussion

Scientific paper

10.1080/00949650802142667

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 gamma is in [-1,0]. Replacing the order statistics in tail index estimators by their corresponding quantiles from the distribution function that is based on the estimated log-concave density leads to novel smooth quantile and tail index estimators. These new estimators aim at estimating the tail index especially 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 well superior to their non-smoothed counterparts, in terms of mean squared error.

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