Abstract
This paper proposes some new estimators for the tail index of a heavy tailed distribution when only a few largest values are observed within blocks. These estimators are proved to be asymptotically normal under suitable conditions, and their Edgeworth expansions are obtained. Empirical likelihood method is also employed to construct confidence intervals for the tail index. The comparison for the confidence intervals based on the normal approximation and the empirical likelihood method is made in terms of coverage probability and length of the confidence intervals. The simulation study shows that the empirical likelihood method outperforms the normal approximation method.
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References
Balakrishnan N. and Cohen A.C. (1991). Order statistics and inference. Academic Press, Boston
Bingham N.H., Goldie C.M. and Teugels J.L. (1987). Regular variation. Cambridge University, New York
Chen S. (1996). Empirical likelihood confidence intervals for nonparametric density estimation. Biometrika 83: 329–341
Chen S. and Hall P. (1993). Smoothed empirical likelihood confidence intervals for quantiles. The Annals of Statistics 21: 1166–1181
Chen S. and Qin Y.S. (2000). Empirical likelihood confidence intervals for local linear smoothers. Biometrika 87: 946–953
Davydov Yu., Paulauskas V. and Račkauskas A. (2000). More on P-stable convex sets in Banach spaces. Journal of Theoretical Probability 13: 39–64
De Haan L., Peng L. (1998). Comparison of tail index estimators. Statistica Neerlandica 32: 60–70
De Haan L. and Stadtmüller U. (1996). Generalized regular variation of second order. Journal of the Australian Mathematical Society (Series A) 61: 381–395
Dekkers A.L.M., Einmahl J.H.J. and de Haan L. (1989). A moment estimator for the index of an extreme-value distribution. The Annals of Statistics 17: 1833–1855
Drees H. (1998). On smooth statistical tail functionals. Scandinavian Journal of Statistics 25: 187–210
Embrechts P., Klüppelberg C. and Mikosch T. (1997). Modelling extremal events for insurance and finance. Springer, Berlin
Gadeikis K. and Paulauskas V. (2005). On the estimation of a changepoint in a tail index. Lithuanian Mathematical Journal 45: 272–283
Geluk J.L., de Haan L. (1987). Regular variation, extensions and Tauberian theorems. CWI Tract, 40. Amsterdam: Centre for Mathematics and Computer Science.
Hall P. and La Scala L. (1990). Methodology and algorithms of empirical likelihood. International Statistical Review 58: 109–127
Hill B.M. (1975). A simple general approach to inference about the tail of a distribution. The Annals of Statistics 3: 1163–1174
Kolaczyk E.D. (1994). Empirical likelihood for generalized linear models. Statistica Sinica 4: 199–218
Lu J. and Peng L. (2002). Likelihood based confidence intervals for the tail index. Extremes 5(4): 337–352
Lu X. and Qi Y. (2004). Empirical likelihood for the additive risk model. Probability and Mathematical Statistics 24: 419–431
Owen A. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75: 237–249
Owen A. (1990). Empirical likelihood regions. The Annals of Statistics 18: 90–120
Owen A. (1991). Empirical likelihood for linear models. The Annals of Statistics 19: 1725–1747
Owen A. (2001). Empirical Likelihood. Chapman and Hall, London
Paulauskas V. (2003). A new estimator for a tail index. Acta Applicandae Mathematicae 79: 55–67
Peng L. and Qi Y. (2004). Estimating the first- and second-order parameters of a heavy-tailed distribution. Australian & New Zealand Journal of Statistics 46: 305–312
Peng L. and Qi Y. (2006a). A new calibration method of constructing empirical likelihood-based confidence intervals for the tail index. Australian & New Zealand Journal of Statistics 48: 59–66
Peng L. and Qi Y. (2006b). Confidence regions for high quantiles of a heavy-tailed distribution. The Annals of Statistics 34(4): 1964–1986
Petrov V.V. (1995). Limit theorems of probability theory: sequences of independent random variables. Clarendon Press, Oxford
Pickands J. (1975). Statistical inference using extreme order statistics. The Annals of Statistics 3: 119–131
Qin G. and Jing B.Y. (2001a). Empirical likelihood for censored linear regression. Scandinavian Journal of Statistics 28: 661–673
Qin G. and Jing B.Y. (2001b). Censored partial linear models and empirical likelihood. Journal of Multivariate Analysis 78: 37–61
Qin G. and Jing B.Y. (2001c). Empirical likelihood for Cox regression model under random censorship. Communications in Statistics Simulation and Computation 30: 79–90
Qin J. and Lawless J. (1994). Empirical likelihood and general estimating equations. The Annals of Statistics 22: 300–325
Segers J. and Teugels J. (2000). Testing the Gumbel hypothesis by the Galton’s ratio. Extremes 3(3): 291–303
Tsao M. (2004). A new method of calibration for the empirical loglikelihood ratio. Statistics & Probability Letters 68: 305–314
Wang Q.H. and Jing B.Y. (2003). Empirical likelihood for partial linear models. Annals of the Institute of Statistical Mathematics 55: 585–595
Wang Q.H. and Li G. (2002). Empirical likelihood semiparametric regression analysis under random censorship. Journal of Multivariate Analysis 83: 469–486
Wang Q.H. and Rao J.N.K. (2002). Empirical likelihood-based inference in linear errors-in-covariables models with validation data. Biometrika 89: 345–358
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The research was supported by NSF grant DMS 0604176.
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Qi, Y. On the tail index of a heavy tailed distribution. Ann Inst Stat Math 62, 277–298 (2010). https://doi.org/10.1007/s10463-008-0176-2
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DOI: https://doi.org/10.1007/s10463-008-0176-2