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Exact tests and confidence sets for the tail coefficient of a-stable distributions


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  • Dufour, Jean-Marie
  • Kurz-Kim, Jeong-Ryeol


In this paper, using the Monte Carlo (MC) method we propose an estimation and (at the same time) a test procedure for the stability parameter of a-stable distributions. One powerful advantage of the MC method is that it provides an exact significance level for finite samples, whose distribution can be far different from that of asymptotic samples on which the level of confidence interval for estimates is usually based. Statistical theory for the MC method is given. A simulation study compares the efficiency of our estimate with the Hill estimate (Hill, 1975). Construction of significance level based on the MC method is exploited and the corresponding power function is also studied. An empirical application demonstrates an easy implementation of our estimation and test procedure. It turns out that our estimate can improve the efficiency of any estimator for a in terms of mean square error. -- In der vorliegenden Arbeit wird ein auf der Monte-Carlo-Methode basierendes Schätz- und Testverfahren für den Stabilitätsparameter von a-stabilen Verteilungen vorgeschlagen. Ein entscheidender Vorteil dieses Verfahrens liegt darin, dass es genauere Konfidenzintervalle für endliche Stichprobenumfänge angibt, die häufig von denen aus asymptotisch ermittelten Verteilungen abweichen. Die statistische Theorie für die Monte-Carlo-Methode wird abgeleitet. Anhand einer Simulationsuntersuchung wird die Effizienz von unserem Verfahren und dem Schätzverfahren von Hill (1975) verglichen. Es wird gezeigt, wie sich die Konfidenzintervalle durch die Monte- Carlo-Methode konstruieren lassen. Zudem werden die zugehörigen Gütefunktionen berechnet. Weiterhin zeigt ein empirisches Beispiel die Einfachheit der empirischen Implementierung unseres Verfahrens. Es wird deutlich, dass unser Verfahren die Effizienz beliebiger Schätzer für a im Sinne vom mittleren quadratischen Fehler verbessern kann.

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Bibliographic Info

Paper provided by Deutsche Bundesbank, Research Centre in its series Discussion Paper Series 1: Economic Studies with number 2003,16.

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Date of creation: 2003
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Handle: RePEc:zbw:bubdp1:4213

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  1. Einmahl, J. & Dekkers, A. & de Haan, L., 1989. "A moment estimator for the index of an extreme-value distribution," Open Access publications from Tilburg University urn:nbn:nl:ui:12-125712, Tilburg University.
  2. Weron, Rafal, 1996. "On the Chambers-Mallows-Stuck method for simulating skewed stable random variables," Statistics & Probability Letters, Elsevier, Elsevier, vol. 28(2), pages 165-171, June.
  3. McCulloch, J Huston, 1997. "Measuring Tail Thickness to Estimate the Stable Index Alpha: A Critique," Journal of Business & Economic Statistics, American Statistical Association, American Statistical Association, vol. 15(1), pages 74-81, January.
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Cited by:
  1. Miller, J. Isaac & Park, Joon Y., 2010. "Nonlinearity, nonstationarity, and thick tails: How they interact to generate persistence in memory," Journal of Econometrics, Elsevier, Elsevier, vol. 155(1), pages 83-89, March.
  2. Miller, J. Isaac & Park, Joon Y., 2005. "How They Interact to Generate Persistency in Memory," Working Papers, Rice University, Department of Economics 2005-01, Rice University, Department of Economics.


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