# zbMATH — the first resource for mathematics

Existence and consistency of the maximum likelihood estimators for the extreme value index within the block maxima framework. (English) Zbl 1388.62042
Summary: The maximum likelihood method offers a standard way to estimate the three parameters of a generalized extreme value (GEV) distribution. Combined with the block maxima method, it is often used in practice to assess the extreme value index and normalization constants of a distribution satisfying a first order extreme value condition, assuming implicitly that the block maxima are exactly GEV distributed. This is unsatisfactory since the GEV distribution is a good approximation of the block maxima distribution only for blocks of large size. The purpose of this paper is to provide a theoretical basis for this methodology. Under a first order extreme value condition only, we prove the existence and consistency of the maximum likelihood estimators for the extreme value index and normalization constants within the framework of the block maxima method.

##### MSC:
 62F10 Point estimation 62G30 Order statistics; empirical distribution functions
AS 215
Full Text:
##### References:
  Balkema, A.A. and de Haan, L. (1974). Residual life time at great age. Ann. Probab. 2 792-804. · Zbl 0295.60014 · doi:10.1214/aop/1176996548  Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes. Theory and Applications. Wiley Series in Probability and Statistics . Chichester: Wiley. · Zbl 1070.62036 · doi:10.1002/0470012382  Billingsley, P. (1995). Probability and Measure , 3rd ed. Wiley Series in Probability and Mathematical Statistics . New York: Wiley. · Zbl 0822.60002  de Haan, L. (1984). Slow variation and characterization of domains of attraction. In Statistical Extremes and Applications ( Vimeiro , 1983). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 131 31-48. Dordrecht: Reidel. · doi:10.1007/978-94-017-3069-3_4  de Haan, L. and Ferreira, A. (2006). Extreme Value Theory. An Introduction. Springer Series in Operations Research and Financial Engineering . New York: Springer. · Zbl 1101.62002  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. · Zbl 0701.62029 · doi:10.1214/aos/1176347397  Drees, H., Ferreira, A. and de Haan, L. (2004). On maximum likelihood estimation of the extreme value index. Ann. Appl. Probab. 14 1179-1201. · Zbl 1102.62051 · doi:10.1214/105051604000000279 · arxiv:math/0407062  Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. For Insurance and Finance. Applications of Mathematics ( New York ) 33 . Berlin: Springer. · Zbl 0873.62116  Hill, B.M. (1975). A simple general approach to inference about the tail of a distribution. Ann. Statist. 3 1163-1174. · Zbl 0323.62033 · doi:10.1214/aos/1176343247  Hosking, J.R.M. (1985). Algorithm AS 215: Maximum likelihood estimation of the parameters of the generalized extreme value distribution. Appl. Stat. 34 301-310.  Hosking, J.R.M., Wallis, J.R. and Wood, E.F. (1985). Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics 27 251-261. · doi:10.1080/00401706.1985.10488049  Macleod, A.J. (1989). AS R76 - A remark on algorithm AS 215: Maximum likelihood estimation of the parameters of the generalized extreme value distribution. Appl. Stat. 38 198-199.  Petrov, V.V. (1995). Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Oxford Studies in Probability 4 . New York: The Clarendon Press Oxford Univ. Press. · Zbl 0826.60001  Pickands, J. III (1975). Statistical inference using extreme order statistics. Ann. Statist. 3 119-131. · Zbl 0312.62038 · doi:10.1214/aos/1176343003  Prescott, P. and Walden, A.T. (1980). Maximum likelihood estimation of the parameters of the generalized extreme-value distribution. Biometrika 67 723-724. · doi:10.1093/biomet/67.3.723  Prescott, P. and Walden, A.T. (1983). Maximum likelihood estimation of the parameters of the three-parameter extreme-value distribution from censored samples. J. Stat. Comput. Simul. 16 241-250. · Zbl 0501.62016 · doi:10.1080/00949658308810625  Shorack, G.R. and Wellner, J.A. (1986). Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics : Probability and Mathematical Statistics . New York: Wiley. · Zbl 1170.62365  Smith, R.L. (1985). Maximum likelihood estimation in a class of nonregular cases. Biometrika 72 67-90. · Zbl 0583.62026 · doi:10.1093/biomet/72.1.67  Smith, R.L. (1987). Estimating tails of probability distributions. Ann. Statist. 15 1174-1207. · Zbl 0642.62022 · doi:10.1214/aos/1176350499  van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3 . Cambridge: Cambridge Univ. Press. · Zbl 0910.62001 · doi:10.1017/CBO9780511802256  Wald, A. (1949). Note on the consistency of the maximum likelihood estimate. Ann. Math. Stat. 20 595-601. · Zbl 0034.22902 · doi:10.1214/aoms/1177729952  Zhou, C. (2009). Existence and consistency of the maximum likelihood estimator for the extreme value index. J. Multivariate Anal. 100 794-815. · Zbl 1169.62038 · doi:10.1016/j.jmva.2008.08.009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.