Abstract
It is known that bootstrapping maximum for estimating the endpoint of a distribution function is inconsistent and subsample bootstrap method is needed. Under an extreme value condition, some other estimators for the endpoint have been studied in the literature, which are preferrable to the maximum in regular cases. In this paper, we show that the full sample bootstrap method is consistent for the endpoint estimator proposed by Hall (1982).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aarssen, K. and de Haan, L. (1994). On the maximal life span of humans. Math. Popul. Stud., 4, 259–281.
Angus, J. (1993). Asymptotic theory for bootstrapping the extremes. Comm. Statist. Theory Methods, 22, 15–30.
Athreya, K.B. and Fukuchi, J.I. (1997). Confidence intervals for endpoints of a c.d.f. via bootstrap. J. Statist. Plann. Inference, 58, 299–320.
Bickel, P.J. and Freedman, D.A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist., 9, 1196–1217.
de Haan, L. and Ferreira, A. (2006). Extreme Value Theory, An Introduction. Springer, New York.
de Haan, L. and Stadtmüller, U. (1996). Generalized regular variation of second order. J. Aust. Math. Soc. A, 61, 381–395.
Deheuvels, P., Mason, D. and Shorack, G. (1993). Some results on the influence of extremes on the bootstrap. Ann. Inst. Henri Poincaré, 29, 83–103.
Dekkers, A.L.M., Einmahl, J.H.J. and de Haan, L. (1989). A moment estimator for the index of an extreme-value distribution. Ann. Statist., 17, 1833–1855.
Drees, H., Ferreira, A. and de Haan, L. (2004). On maximum likelihood estimation of the extreme value index. Ann. Appl. Probab., 14, 1179–1201.
Ferreira, A., de Haan, L. and Peng, L. (2003). On optimising the estimation of high quantiles of a probability distribution. Statistics, 37, 403–434.
Fraga Alves, M.I., de Haan, L. and Lin, T. (2003). Estimation of the parameter controlling the speed of convergence in extreme value theory. Math. Methods Statist., 12, 155–176.
Geluk, J. and de Haan, L. (2002). On bootstrap sample size in extreme value theory. Publ. Inst. Math. (Beograd) (N.S.), 71, 21–25.
Hall, P. (1982). On estimating the endpoint of a distribution. Ann. Statist., 10, 556–568.
Hall, P. and Wang, J.Z. (1999). Estimating the end-point of a probability distribution using minimum-distance methods. Bernoulli, 5, 177–189.
Li, D. and Peng, L. (2009). Does Bias Reduction with External Estimator of Second Order Parameter Work for endpoint? J. Statist. Plann. Inference, 139, 1937–1952.
Li, D. and Peng, L. (2010). Compare extreme models when the sign of the extreme value index is known. Statist. Probab. Lett., 80, 739–746.
Loh, W.Y. (1984). Estimating an endpoint of a distribution with resampling methods. Ann. Statist., 12, 1534–1550.
Peng, L. and Qi, Y. (2009). Maximum likelihood estimation of extreme value index for irregular cases. J. Statist. Plann. Inference, 139, 3361–3376.
Qi, Y. (2008). Bootstrap and empirical likelihood methods in extremes. Extremes, 11, 81–97.
Smith, R.L. (1987). Estimating tails of probability distributions. Ann. Statist., 15, 1174–1207.
Swanepoel, J.W.H. (1986). A note on proving that the (modified) bootstrap works. Comm. Statist. Theory Methods, 15, 3193–3203.
Woodroofe, M. (1974). Maximum likelihood estimation of translation parameter of truncated distribution II. Ann. Statist., 2, 474–488.
Acknowledgement
We thank a reviewer and a co-editor for helpful comments. Peng’s research was supported by NSA grant H98230-10-1-0170 and NSF grant DMS-1005336.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, Z., Peng, L. Bootstrapping endpoint. Sankhya A 74, 126–140 (2012). https://doi.org/10.1007/s13171-012-0015-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13171-012-0015-7