Estimating the parameters of rare events. (English) Zbl 0722.62021

Hsing, Tailen

Stochastic Processes Appl. 37, No. 1, 117-139 (1991).

In the i.i.d. case the famous limit theorem for the k-th maximum of n random variables plays an important role in the modeling of extremes. The author considers a special dependent stationary sequence of random variables, and observes a clustering of indices belonging to extreme values. A brief introduction to extremal theory for dependent sequences and a corresponding limit theorem lead to the statistical problems. To estimate the parameters (e.g. the extremal index) a sample is divided into a number of blocks; thus the question arises, how the number of sub- samples (blocks) and a high level should vary with the sample size. In the sequel, weak consistency of the proposed estimators is examined in a general framework, and results on strong consistency are given under more restrictive conditions.
The last section is devoted to estimation and asymptotic properties. Using the author’s words it is “the first attempt to understand the problem of estimating the extremal index and related parameters”. In view of encouraging results further efforts on this topic should be done. Possibly the assumptions used in the present article can be modified or simplified to make their verification easier, and to get a deeper insight into the structure of the problems. Some suggestions for ensuing examinations are stated at the end of this interesting paper.

Reviewer: B.Rauhut (Aachen)

MSC:

62F12	Asymptotic properties of parametric estimators
62M99	Inference from stochastic processes
62G30	Order statistics; empirical distribution functions

Keywords:

tail probability; extreme order statistics; modeling of extremes; dependent stationary sequence of random variables; clustering of indices; limit theorem; extremal index; weak consistency; strong consistency

References:

[1]	Billingsley, P., Convergence of probability measures, (1968), Wiley New York · Zbl 0172.21201
[2]	Davis, R.A.; Resnick, S.I., Tail estimates motivated by extreme value theory, Ann. statist., 12, 1467-1487, (1984) · Zbl 0555.62035
[3]	Davis, R.A.; Resnick, S.I., Limit theory for moving averages of random variables with regularly varying tail probabilities, Ann. probab., 13, 179-195, (1985) · Zbl 0562.60026
[4]	Feller, W., An introduction to probability theory and its applications, Vol. 2, (1971), Wiley New York · Zbl 0219.60003
[5]	Gnedenko, B.V., Sur la distribution limit du terme maximum d’une séries aléatoire, Ann. math., 44, 412-453, (1943) · Zbl 0063.01643
[6]	Gumbel, E.J., Statistics of extremes, (1958), Columbia Univ. Press New York · Zbl 0086.34401
[7]	de Haan, L., On regular variation and its application to the weak convergence of sample extremes, Amsterdam math. centre tracts, 32, (1970), Math. Centre Amsterdam · Zbl 0226.60039
[8]	Hoeffding, W., Probability inequalities for sums of bounded random variables, J. amer. statist. assoc., 58, 13-30, (1963) · Zbl 0127.10602
[9]	Hsing, T., On the weak convergence of extreme order statistics, Stochastic process. appl., 29, 155-169, (1988) · Zbl 0654.60021
[10]	Hsing, T., On tail index estimation using dependent data, (1989), Preprint
[11]	Hsing, T.; Hüsler, J.; Leadbetter, M.R., On the exceedance point process for a stationary sequence, Probab. theory rel. fields, 78, 97-112, (1988) · Zbl 0619.60054
[12]	Ibragimov, I.A.; Linnik, Y.V., Independent and stationary sequences of random variables, (1969), Wolters-Noordhoff Groningen
[13]	Leadbetter, M.R., On extreme values in stationary sequences, Z. wahrsch. verw. gebiete, 28, 289-303, (1974) · Zbl 0265.60019
[14]	Leadbetter, M.R., Extremes and local dependence in a stationary sequence, Z. wahrsch. verw. gebiete, 65, 291-306, (1983) · Zbl 0506.60030
[15]	Leadbetter, M.R.; Lindgren, G.; Rootzén, H., Extremes and related properties of random sequences and processes, (1983), Springer New York · Zbl 0518.60021
[16]	Leadbetter, M.R.; Weissman, I.; de Haan, L.; Rootzén, H., On clustering of high values in statistically stationary series, Tech. rept. no. 253, (1989), Center for Stochastic Processes, Statist. Dept., Univ. of North Carolina Chapel Hill, NC
[17]	Pickands, J., Statistical inference using extreme order statistics, Ann. statist., 3, 119-131, (1975) · Zbl 0312.62038
[18]	Smith, R.L., Estimating tails of probability distributions, Ann. statist., 15, 1174-1207, (1987) · Zbl 0642.62022
[19]	Volkonskii, V.A.; Rozanov, Y.A., Some limit theorems for random functions, I, Theory probab. appl., 4, 178-197, (1959) · Zbl 0092.33502
[20]	Weissman, I., Estimation of parameters and large quantiles based on k largest observations, J. amer. statist. assoc., 73, 812-815, (1978) · Zbl 0397.62034