Small Sample Estimation and Testing for Heavy Tails
Milan Stehlík
Department of Applied Statistics, Johannes Kepler University in Linz, Linz, Austria

If the model underlying the data, is a regularly varying function with index -1/alpha it is usually supposed that the top scaled order statistics are Pareto distributed. Hill (1975) derived a procedure of Pareto tail estimation by the MLE. The influence function of Hill estimator is slowly increasing, but unbounded. Hill procedure is thus no robust and many authors tried to make the original Hill robust. In Fabián (2001) a new method of score moment estimators has been proposed. It appeared that these score moment estimators are robust for a heavy tailed distributions (see Stehlík et al. (2010)). In Fabián and Stehlík (2010) we understand “The Hill estimator” as a specific procedure for studying of the Pareto tail. Instead of implementing “The Hill estimator” procedure, we implement the score moment procedure. For the case of Pareto distribution, the Hill estimator procedure with the score moment estimator has been investigated in Stehlík et al. (2011) for optimal testing for normality against Pareto tail. In literature, mainly the asymptotical properties of tail estimators are studied. However, in many situations, asymptotics is simplifying the underlying process too much. We may illustrate this fact by a severe bias of the Hill based estimators or by a distributional insensitivity of asymptotical estimators. In many practical situations the robust estimator with good small sample properties is needed. We will show how to construct a robust, distribution sensitive heavy tail estimator and prove the asymptotic normality, weak and strong consistency together with its good small sample properties.


Fabián Z. (2001). Induced cores and their use in robust parametric estimation, Communication in Statistics, Theory Methods, 30: 537-556. Fabián Z. and Stehlík M. (2010). On robust and distribution sensitive Hill like method, Tech. report

Stehlík M., Potocký R., Waldl H. and Fabián Z. (2010). On the favourable estimation of fitting heavy tailed data, Computational Statistics, 25:485-503.

Stehlík M., Fabián Z. and Strelec L. (2011). Small sample robust testing for Normality against Pareto tails, JSPI, accepted.

Keywords: Hill estimator; Heavy tail estimation; Robustness; Testing

Biography: Dr. Milan Stehlik has finished his PhD in Statistics in 2003 at Comenius University, Bratislava, Slovakia.

Later he was involved in several international projects and collaborations in Austria, Spain, Russia, Canada, Germany among others. Currently he is assistant professor for Professor W.G.

Mueller at Department of Statistics, Johannes Kepler University in Linz, with whom he collaborates since 2003. He is active mainly in the experimental design, exact testing, life data modelling and reliability theory.