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A PARAMETRIC BOOTSTRAP FOR HEAVY-TAILED DISTRIBUTIONS

Published online by Cambridge University Press:  08 September 2014

Adriana Cornea-Madeira  and
Russell Davidson
Adriana Cornea-Madeira*
Affiliation:
University of York
Russell Davidson
Affiliation:
McGill University
*
*Address correspondence to Adriana Cornea-Madeira, University of York, Freboys Lane, Heslington, York YO10 5GD, United Kingdom; e-mail: adriana.cornea-madeira@york.ac.uk or to Russell Davidson, McGill University, Department of Economics and CIREQ, H3A 2T7 Montréal, Quebec, Canada; e-mail: russell.davidson@mcgill.ca.
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Abstract

It is known that Efron’s bootstrap of the mean of a distribution in the domain of attraction of the stable laws with infinite variance is not consistent, in the sense that the limiting distribution of the bootstrap mean is not the same as the limiting distribution of the mean from the real sample. Moreover, the limiting bootstrap distribution is random and unknown. The conventional remedy for this problem, at least asymptotically, is either the m out of n bootstrap or subsampling. However, we show that both these procedures can be unreliable in other than very large samples. We introduce a parametric bootstrap that overcomes the failure of Efron’s bootstrap and performs better than the m out of n bootstrap and subsampling. The quality of inference based on the parametric bootstrap is examined in a simulation study, and is found to be satisfactory with heavy-tailed distributions unless the tail index is close to 1 and the distribution is heavily skewed.

Type
ARTICLES
Information
Econometric Theory , Volume 31 , Issue 3 , June 2015 , pp. 449 - 470
Copyright
Copyright © Cambridge University Press 2014 

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Footnotes

We thank Michael Jansson, Keith Knight, Peter C.B. Phillips, and the referees for their helpful suggestions. This research was supported by the British Academy’s PDF/2009/370, the Canada Research Chair program (Chair in Economics, McGill University) and by grants from the Social Sciences and Humanities Research Council of Canada, and the Fonds Québécois de Recherche sur la Société et la Culture.

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