Small sample estimation and testing for heavy tails

0390574 - ÚI 2013 RIV NL eng C - Conference Paper (international conference)
Fabián, Zdeněk - Stehlík, M. - Střelec, L.
Small sample estimation and testing for heavy tails.
Proceedings of the 58th World Statistics Congress 2011. The Hague: International Statistical Institute, 2012, s. 2985-2993. ISBN 978-90-73592-33-9.
[ISI 2011. Session of the International Statistical Institute /58./. Dublin (IE), 21.08.2011-26.08.2011]
Institutional research plan: CEZ:AV0Z10300504
Keywords : Hill estimator * heavy tail estimation * robustness * testing
Subject RIV: BB - Applied Statistics, Operational Research
http://2011.isiproceedings.org/papers/650188.pdf

The aim of the paper is to introduce the distribution sensitive tail estimation procedure, which is easy to implement for various distributions. Here we construct distribution-sensitive estimators based on the Hills procedure using score moment estimators for other heavy-tailed distributions (Pareto, Fréchet, Burr, log-gama, inverse gamma, etc.), and to study their statistical properties, both theoretically (e.g consistency, asymptotical distribution) and by means of simulation experiments (e.g. comparisons of exact effectiveness of our method for different heavy tailed distributions). A specific problem is finding of an optimal threshold k, say, yielding a trade off in between of variance and bias of the Hill estimator. Simulation results by Embrechts et al. (1997) showed that the Hill estimator and its alternatives work well over large ranges of values for k in the case of Pareto distribution. However, Hill estimator is often giving wrong results for distributions different from the Pareto one. Their "Hill horror plots" actually show deviations of the Hill estimates trending farther away from the true value of the tail index as k is increased. In this paper we illustrate t-Hill. We also quantify the robustness and compare efficiency with other competitors. The paper is organized as follows. First section is introduction. In section 2 we recall the theory of scalar score. In section 3 we discussed the t-Hill estimator, introduced firstly in Stehlík et al. (2011). The section 4 comparing t-Hill and Hill estimators follows. Therein contamination of underling data is controlled by means of score variance of Pareto distribution. Comparisons show that t-Hill estimator outperforms Hill estimator. In section 5 we introduce the t-Hill plot. We end with powers of selected tests for normality against Pareto distribution.
Permanent Link: http://hdl.handle.net/11104/0219440