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

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
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