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A statistical application of the quantile mechanics approach: MTM estimators for the parameters of t and gamma distributions

Published online by Cambridge University Press:  18 May 2012

A. KLEEFELD
Affiliation:
Department I (Mathematics, Natural Sciences, Computer Science), Brandenburgische Technische Universität, P.O. Box 101344, 03013 Cottbus, Germany email: kleefeld@tu-cottbus.de
V. BRAZAUSKAS
Affiliation:
Department of Mathematical Sciences, University of Wisconsin – Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA email: vytaras@uwm.edu

Abstract

In this paper, we revisit the quantile mechanics approach, which was introduced by Steinbrecher and Shaw (Steinbrecher, G. & Shaw, W. T. (2008) Quantile mechanics. European. J. Appl. Math.19, 87–112). Our objectives are (i) to derive the method of trimmed moments (MTM) estimators for the parameters of gamma and Student's t distributions, and (ii) to examine their large- and small-sample statistical properties. Since trimmed moments are defined through the quantile function of the distribution, quantile mechanics seems like a natural approach for achieving objective (i). To accomplish the second goal, we rely on the general large sample results for MTMs, which were established by Brazauskas et al. (Brazauskas, V., Jones, B. L. & Zitikis, R. (2009) Robust fitting of claim severity distributions and the method of trimmed moments. J. Stat. Plan. Inference139, 2028–2043), and then use Monte Carlo simulations to investigate small-sample behaviour of the newly derived estimators. We find that, unlike the maximum likelihood method, which usually yields fully efficient but non-robust estimators, the MTM estimators are robust and offer competitive trade-offs between robustness and efficiency. These properties are essential when one employs gamma or Student's t distributions in such outlier-prone areas as insurance and finance.

Type
Papers
Copyright
Copyright © Cambridge University Press 2012

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