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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.

References

REFERENCES

Arcones, M.A. & Giné, E. (1989) The bootstrap of the mean with arbitrary bootstrap sample size. Annales de l’Institut Henri Poincaré 25(4), 457–481.Google Scholar
Arlitt, M.F. & Williamson, C.L. (1996) Web server workload characterization: The search for invariants. In: Proceedings of the 1996 ACM SIGMETRICS Conference on Measurement and Modeling of Computer Systems, pp. 126–137.CrossRefGoogle Scholar
Athreya, K.B. (1985) Bootstrap for the Mean in the Infinite Variance Case, II. Technical report 86-21, Iowa State University, Department of Statistics.Google Scholar
Athreya, K.B. (1987) Bootstrap of the mean in the infinite variance case. Annals of Statistics 15, 724–731.CrossRefGoogle Scholar
Beran, R. (1988) Prepivoting test statistics: A bootstrap view of asymptotic refinements. Journal of the American Statistical Association 83, 307–327.CrossRefGoogle Scholar
Bertail, P., Politis, D.N., & Romano, J.P. (1999) On subsampling estimators with unknown rate of convergence. Journal of the American Statistical Association 94, 569–579.CrossRefGoogle Scholar
Bickel, P.J., Götze, F., & van Zwet, W.R. (1997) Resampling fewer than n observations: Gains, loses and remedies for losses. Statistica Sinica 7, 1–31.Google Scholar
Bickel, P.J. & Sakov, A. (2008) On the choice of m in the m out of n bootstrap and confidence bounds for extrema. Statistica Sinica 18, 967–985.Google Scholar
Bingham, N.H., Goldie, C.M., & Teugles, J. (1989) Regular Variation. Cambridge University Press.Google Scholar
Bodnarchuk, R.R. & Bunt, R.B. (1991) A synthetic workload model for a distributed system file server. In: Proceedings of the 1991 ACM SIGMETRICS Conference on Measurement and Modeling of Computer Systems, pp. 50–59.CrossRefGoogle Scholar
Chambers, J.M., Mallows, C., & Stuck, C. (1981) A method for simulating stable random variables. Journal of the American Statistical Association 71, 340–344.CrossRefGoogle Scholar
Cowell, F.A. (1995) Measuring Inequality, 2nd ed. Harvester Wheatsheaf.Google Scholar
Danielsson, J., de Haan, L., Peng, L., & de Vries, C.G. (2001) Using a bootstrap method to choose the sample fraction in tail index estimation. Journal of Multivariate Analysis 76, 226–248.CrossRefGoogle Scholar
Davis, R. & Resnick, S. (1984) Tail estimates motivated by extreme value theory. Annals of Statistics 12, 1467–1487.CrossRefGoogle Scholar
Efron, B. (1969) Student’s t-test under symmetry conditions. Journal of the American Statistical Association 64, 1278–1302.Google Scholar
Efron, B. (1979) Bootstrap methods: Another look at the jackknife. Annals of Statistics 7, 1–26.CrossRefGoogle Scholar
Embrechts, P. (2001) Extremes in economics and the economics of extremes. In Finkenstadt, B. & Rootzen, H. (eds.), Extreme Values in Finance, Telecommunications and the Environment 7, pp. 169–183. Chapman and Hall/CRC.Google Scholar
Embrechts, P., Kluppelberg, C., & Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance. Springer-Verlag.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, vol. II.Wiley.Google Scholar
Fiorio, C.V., Hajivassiliou, V., & Phillips, P.C.B. (2010) Bimodal t-ratios: The impact of thick tails on inference. Econometrics Journal 13(2), 271–289.CrossRefGoogle Scholar
Gabaix, X. (1999) Zipf’s law for cities: An explanation. Quarterly Journal of Economics 114, 739–767.CrossRefGoogle Scholar
Giné, E., Götze, F., & Mason, D.M. (1997) When is the Student t-statistic asymptotically standard normal? Annals of Probability 25, 1514–1531.Google Scholar
Gnedenko, B.V. & Kolmogorov, A.N. (1954) Limit Distributions for Sums of Independent Random Variables. Addison-Wesley.Google Scholar
Haeusler, E. & Teugles, J.L. (1985) On asymptotic normality of Hill’s estimator for the exponent of regular variation. Annals of Statistics 13, 743–756.CrossRefGoogle Scholar
Hall, P. (1982) On some simple estimates of an exponent of regular variation. Journal of the Royal Statistical Society, Series B 44, 37–42.Google Scholar
Hall, P. (1990a) Asymptotic properties of the bootstrap for heavy-tailed distributions. Annals of Probability 18(2), 1342–1360.CrossRefGoogle Scholar
Hall, P. (1990b) Using the bootstrap to estimate mean squared error and select smoothing parameters in nonparametric problems. Journal of Multivariate Analysis 32, 177–203.CrossRefGoogle Scholar
Hall, P. & Jing, B. (1998) Comparison of the bootstrap and asymptotic approximations to the distribution of a heavy-tailed mean. Statistica Sinica 8, 887–906.Google Scholar
Hall, P. & Yao, Q. (2003) Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71(1), 285–317.CrossRefGoogle Scholar
Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution. Annals of Statistics 3, 1163–1174.CrossRefGoogle Scholar
Ibragimov, R. (2007) Efficiency of linear estimators under heavy-tailedness: Convolutions of α-symmetric distributions. Econometric Theory 23, 501–517.CrossRefGoogle Scholar
Ibragimov, R. (2014) On the robustness of location estimators in models of firm growth under heavy-tailedness. Journal of Econometrics 181, 25–33.Google Scholar
Knight, K. (1989) On the bootstrap of the sample mean in the infinite variance case. Annals of Statistics 17, 1168–1175.CrossRefGoogle Scholar
Lévy, P. (1925) Calcul des Probabilités. Gauthier-Villars.Google Scholar
Logan, B.F., Mallows, C., Rice, S., & Shepp, L.A. (1973) Limit distributions of self-normalized sums. Annals of Probability 11, 788–809.CrossRefGoogle Scholar
Loretan, M. & Phillips, P.C.B. (1994) Testing the covariance stationarity of heavy-tailed time series. Journal of Empirical Finance 1(2), 211–248.Google Scholar
Mandelbrot, B. (1963) The variation of other speculative prices. Journal of Business 40, 393–413.CrossRefGoogle Scholar
Mason, D.M. (1982) Laws of large numbers for sums of extreme values. Annals of Probability 10(3), 754–762.Google Scholar
Mittnik, S. & Rachev, S. (2000) Stable Paretian Models in Finance. Wiley.Google Scholar
Mittnik, S., Rachev, S., & Kim, J.R. (1998) Chi-square-type distributions for heavy-tailed variates. Econometric Theory 14, 339–354.Google Scholar
Politis, D.N., Romano, J.P., & Wolf, M. (1999) Subsampling. Springer.Google Scholar
Romano, J.P. & Wolf, M. (1999) Subsampling inference for the mean in the heavy-tailed case. Metrika 50, 55–69.CrossRefGoogle Scholar
Samorodnitsky, G. & Taqqu, M. (1994) Stable Non-Gaussian Random Processes. Chapman and Hall.Google Scholar
Serfling, R.J. (1980) Approximation Theorems of Mathematical Statistics. Wiley.CrossRefGoogle Scholar
Shao, J. & Tu, D. (1995) The Jackknife and Bootstrap. Springer-Verlag.CrossRefGoogle Scholar
Steindl, J. (1965) Random Processes and the Growth of Firms. Griffin.Google Scholar
Zipf, G. (1949) Human Behavior and the Principle of the Last Effort. Addison-Wesley.Google Scholar
Zolotarev, V.M. (1986) One-dimensional Stable Distributions. Translations of Mathematical Monographs, vol. 65. American Mathematical Society.CrossRefGoogle Scholar