Abstract
When modeling extreme events, there are a few primordial parameters, among which we refer to the extreme value index (EVI) and the extremal index (EI). Under a framework related to large values, the EVI measures the right tail weight of the underlying distribution and the EI characterizes the degree of local dependence in the extremes of a stationary sequence. Most of the semiparametric estimators of these parameters show the same type of behavior: nice asymptotic properties but a high variance for small values of k, the number of upper order statistics used in the estimation, and a high bias for large values of k. This brings a real need for the choice of k. Choosing some well-known estimators of those two parameters, we revisit the application of a heuristic algorithm for the adaptive choice of k. A simulation study illustrates the performance of the proposed algorithm.
Similar content being viewed by others
References
Alpuim, M. T. 1989. An extremal markovian sequence. J. Appl. Prob., 26, 219–222.
Beirlant, J., Y. Goegebeur, J. Segers, J. Teugels, D. Waal, and C. Ferro. 2004. Statistics of extremes: Theory and applications. New York, NY: John Wiley & Sons.
Bingham, N. H., C. M. Goldie, and J. L. Teugels. 1987. Regular variation. New York, NY: Cambridge University Press.
Caeiro, F., M. I. Gomes, and D. D. Pestana. 2005. Direct reduction of bias of the classical Hill estimator. Revstat 3(2), 113–136.
Caeiro, F., M. I. Gomes, and L. Henriques-Rodrigues. 2009. Reduced-bias tail index estimators under a third order framework. Commun. Stat. Theory Methods, 38(7), 1019–1040.
Davidson, A. 2011. Statistics of extremes. Courses 2011–2012. École Polytechnique Fédérale de Lausanne EPFL, Lausanne, Switzerland.
Fraga Alves, M. I., M. I. Gomes, and L. de Haan. 2003. A new class of semi-parametric estimators of the second order parameter. Port. Math., 60(2), 194–213.
Gomes, M. I. 1990. Statistical inference in an extremal Markovian model. In COMPSTAT 1990: Proceedings in Computational Statistics, eds. K. Momirovic and V. Mildner, 257–262. Heidelberg: Physica-Verlag.
Gomes, M. I. 1993a. Modelos extremais em esquemas de dependência. In Estatística Robusta, Extremos e Mais Alguns Temas, ed. D. Pestana, 209–220. Lisboa: Salamandra.
Gomes, M. I. 1993b. On the estimation of parameters of rare events in environmental time series. In Statistics for the environment, eds. V. Barnett and K. F. Turkman, 226–241. New York, NY: John Wiley & Sons.
Gomes, M. I., and M. J. Martins. 2002. “Asymptotically unbiased” estimators of the tail index based on external estimation of the second order parameter. Extremes 5(1), 5–31.
Gomes, M. I., L. de Haan, and L. Henriques-Rodrigues. 2008a. Tail index estimation for heavy-tailed models: accommodation of bias in weighted log-excesses. J. R. Stat. Soc., B70(1), 31–52.
Gomes, M. I., A. Hall, and C. Miranda. 2008b. Subsampling techniques and the jackknife methodology in the estimation of the extremal index. J. Comput. Stat. Data Anal., 52(4), 2022–2041.
Gomes, M. I., F. Figueiredo, and M. M. Neves. 2012. Adaptive estimation of heavy right tails: resampling-based methods in action. Extremes, 15, 463–489.
Gomes, M. I., M. J. Martins, and M. M. Neves. 2013. Generalised jackknife-based estimators for univariate extreme-value modelling. Commun. Stat. Theory Methods, 42(7), 1227–1245.
Gray, H. L., and W. R. Schucany. 1972. The generalized jackknife statistic. New York, NY: Marcel Dekker.
Hill, B. 1975. A simple general approach to inference about the tail of a distribution. Ann. Stat., 3, 1163–1174.
Hsing, T., J. Husler, and M. R. Leadbetter. 1988. On exceedance point process for a stationary sequence. Probab. Theory Related Fields, 78, 97–112.
Leadbetter, M. R. 1983. Extremes and local dependence in stationary sequences. Z. Wahrsch. Verw. Gebiete, 65(2), 291–306.
Leadbetter, M. R., G. Lindgren, and H. Rootzén. 1983. Extremes and related properties of random sequences and series. New York, NY: Springer-Verlag.
Leadbetter, M. R., and L. Nandagopalan. 1989. On exceedance point process for stationary sequences under mild oscillation restrictions. In Extreme value theory: Proceedings, Oberwolfach 1987, ed. J. Hüsler and R. D. Reiss, Lecture Notes in Statistics 52, 69–80. Berlin, Germany: Springer-Verlag.
Martins, A. P., and H. Ferreira. 2004. The extremal index of sub-sampled processes. J. Stat. Plan. Inference, 124, 145–152.
Nandagopalan, S. 1990. Multivariate extremes and estimation of the extremal index. PhD thesis, University of North Carolina, Chapel Hill, NC.
Nandagopalan, S., and H. Rootzén. 1988. Extremal theory for stochastic processes. Ann. Probab., 16(2), 431–478.
O’Brien, G. 1987. Extreme values for stationary and Markov sequences. Ann. Probab., 15(1), 281–289.
Prata Gomes, D., and M. M. Neves. 2011. Resampling methodologies and the estimation of parameters of rare events. In Numerical analysis and applied mathematics (ICNAAM 2011), AIP Conf. Proc., 1389, 1475–1478.
Robinson, M. E., and J. A. Tawn. 2000. Extremal analysis of processes sampled at different frequencies. J. R. Stat. Soc. B, 62, 117–135.
Scotto, M., K. F. Turkman, and C. W. Anderson. 2003. Extremes of some sub-sampled time series. J. Time Series Anal., 24(5), 505–512.
Author information
Authors and Affiliations
Corresponding author
Additional information
Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/ujsp.
Rights and permissions
About this article
Cite this article
Neves, M.M., Gomes, M.I., Figueiredo, F. et al. Modeling Extreme Events: Sample Fraction Adaptive Choice in Parameter Estimation. J Stat Theory Pract 9, 184–199 (2015). https://doi.org/10.1080/15598608.2014.890984
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1080/15598608.2014.890984