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Weighted approximations of tail processes for \(\beta\)-mixing random variables. (English) Zbl 1073.60520

Summary: While the extreme value statistics for i.i.d data is well developed, much less is known about the asymptotic behavior of statistical procedures in the presence of dependence.We establish convergence of tail empirical processes to Gaussian limits for \(\beta\)-mixing stationary time series. As a consequence, one obtains weighted approximations of the tail empirical quantile function that is based on a random sequence with marginal distribution belonging to the domain of attraction of an extreme value distribution. Moreover, the asymptotic normality is concluded for a large class of estimators of the extreme value index. These results are applied to stationary solutions of a general stochastic difference equation.

MSC:

60F17 Functional limit theorems; invariance principles
60G10 Stationary stochastic processes
60G70 Extreme value theory; extremal stochastic processes
62G20 Asymptotic properties of nonparametric inference
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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[1] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[2] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press. · Zbl 0617.26001
[3] Doukhan, P. (1995). Mixing. Properties and Examples. Springer, New York. Drees, H. (1998a). On smooth statistical tail functionals. Scand. J. Statist 25 187-210. Drees, H. (1998b). A general class of estimators of the extreme value index. J. Statist. Plann. Inference 66 95-112. Drees, H. (1998c). Estimating the Extreme Value Index. Habilitation thesis, Univ. Cologne. Available at euklid.mi.uni-koeln.de/ hdrees/habil.ps
[4] Drees, H. and de Haan, L. (1999). Conditions for quantile process approximations. Comm. Statist. Stochastic Models 15 485-502. · Zbl 0937.60019 · doi:10.1080/15326349908807546
[5] Drost, F. C. and Klaassen, C. A. J. (1997). Efficient estimationinsemiparametric GARCH models. J. Econometrics 81 193-221. · Zbl 0944.62120 · doi:10.1016/S0304-4076(97)00042-0
[6] Drost, F. C., Klaassen, C. A. J. and Werker, B. J. M. (1997). Adaptive estimationintime-series models. Ann. Statist. 25 786-817. · Zbl 0941.62093 · doi:10.1214/aos/1031833674
[7] Eberlein, E. (1984). Weak convergence of partial sums of absolutely regular sequences. Statist. Probab. Lett. 2 291-293. · Zbl 0564.60025 · doi:10.1016/0167-7152(84)90067-1
[8] Embrechts, P., Kl üppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin. · Zbl 0873.62116
[9] Engle, R. (1982). Autoregressive conditional heteroskedastic models with estimates of the variance of United Kingdom inflation. Econometrica 50 987-1007. JSTOR: · Zbl 0491.62099 · doi:10.2307/1912773
[10] Fernique, X. M. (1975). Regularite des trajectoires des fonctions aleatoires gaussiennes. Ecole d’Eté de Probabilités des Saint-Flour IV. Lecture Notes in Math. 1-96. Springer, Berlin. · Zbl 0331.60025
[11] Gill, R. D. (1989). Nonand semi-parametric maximum likelihood estimators and the von Mises method I. Scand. J. Statist. 16 97-128. · Zbl 0688.62026
[12] Goldie, C. M. (1989). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126-166. · Zbl 0724.60076 · doi:10.1214/aoap/1177005985
[13] 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
[14] Hsing, T. (1991). On tail index estimation using dependent data. Ann. Statist. 19 1547-1569. · Zbl 0738.62026 · doi:10.1214/aos/1176348261
[15] Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207-248. · Zbl 0291.60029 · doi:10.1007/BF02392040
[16] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York. · Zbl 0518.60021
[17] Moore, D. S. and Spruill, M. C. (1975). Unified large-sample theory for general chi-squared statistics for tests of fit. Ann. Statist. 3 599-616. · Zbl 0322.62047
[18] M óricz, F. (1982). A general moment inequality for the maximum of partial sums of single series. Acta Sci. Math. Szeged 44 67-75. · Zbl 0487.60025
[19] Pickands III, J. (1975). Statistical inference using extreme order statistics. Ann. Statist. 3 119-131. · Zbl 0312.62038 · doi:10.1214/aos/1176343003
[20] Puri, M. L. and Tran, L. T. (1980). Empirical distribution functions and functions of order statistics for mixing random variables. J. Multivariate Anal. 10 402-425. · Zbl 0466.62043 · doi:10.1016/0047-259X(80)90061-5
[21] Reiss, R.-D. and Thomas, M. (1997). Statistical Analysis of Extreme Values. Birkhäuser, Basel. · Zbl 0880.62002
[22] Resnick, S. and St aric a, C. (1997). Asymptotic behavior of Hill’s estimator for autoregressive data. Comm. Statist. Stochastic Models 13 703-721. · Zbl 1047.62079 · doi:10.1080/15326349708807448
[23] Resnick, S. and St aric a, C. (1998). Tail index estimation for dependent data. Ann. Appl. Probab. 8 1156-1183. · Zbl 0942.60037 · doi:10.1214/aoap/1028903376
[24] Rootzén, H. (1995). The tail empirical process for stationary sequences. Preprint, Chalmers Univ. Gothenburg. Available at www.math.chalmers.se/Stat/Research/Preprints/ ms95-hr.ps URL:
[25] Rootzén, H. (1998). Example which satisfies (1.1), (2.2), and (2.4) of ”The tail empirical process”, but not the conclusion of Theorem 2.1. Unpublished manuscript.
[26] Shao, Q. M. (1993). Almost sure invariance principles for mixing sequences of random variables. Stochastic Process. Appl. 48 319-334. · Zbl 0793.60038 · doi:10.1016/0304-4149(93)90051-5
[27] Shao, Q. M. and Yu, H. (1996). Weak convergence for weighted empirical processes of dependent sequences. Ann. Probab. 24 2098-2127. · Zbl 0874.60006 · doi:10.1214/aop/1041903220
[28] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York. · Zbl 1170.62365
[29] Smith, R. L. (1987). Estimating tails of probability distributions. Ann. Statist. 15 1174-1207. · Zbl 0642.62022 · doi:10.1214/aos/1176350499
[30] St aric a, C. (1999). On the tail empirical process of solutions of stochastic difference equations. Preprint, Chalmers Univ. Gothenburg. Available at www.math.chalmers.se/ starica/ resume/aarch.ps.gz. URL:
[31] Vervaat, W. (1972). Functional centeral limit theorems for processes with positive drift and their inverses. Z. Wahrsch. Verw. Gebiete 23 245-253. · Zbl 0238.60018 · doi:10.1007/BF00532510
[32] Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Probab. 11 750-783. JSTOR: · Zbl 0417.60073 · doi:10.2307/1426858
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