Abstract
In this paper, we study the existence and consistency of the maximum likelihood estimator of the extreme value index based on k-record values. Following the method used by Drees et al. (2004) and Zhou (2009), we prove that the likelihood equations, in terms of k-record values, eventually admit a strongly consistent solution without any restriction on the extreme value index, which is not the case in the aforementioned studies.
Highlights
Let X1, X2, . . ., be a sequence of independent and identically distributed random variables (i.i.d.) having a continuous distribution function F
Assume that F belongs to the max-domain of attraction of an extreme value distribution, denoted by F ∈ D(Gc) with c ∈ R, i.e., there exist sequences an > 0 and bn ∈ R such that lim n⟶∞
The theory of record values is connected very closely to the extreme value theory through, like, for example, Resnick’s duality theorem or the characterization of tail distributions (e.g., [10]). ere are quite few publications which are devoted to the estimation of the extreme value index based on record values, see, for Journal of Probability and Statistics example, Berred [11], Khaled et al [12], and El Arrouchi and Imlahi [13]
Summary
Let X1, X2, . . ., be a sequence of independent and identically distributed random variables (i.i.d.) having a continuous distribution function F. Assume that F belongs to the max-domain of attraction of an extreme value distribution, denoted by F ∈ D(Gc) with c ∈ R, i.e., there exist sequences an > 0 and bn ∈ R such that lim n⟶∞. Ere are quite few publications which are devoted to the estimation of the extreme value index based on record values, see, for Journal of Probability and Statistics example, Berred [11], Khaled et al [12], and El Arrouchi and Imlahi [13]. We intend to investigate this problem in this paper, so we are interested here to propose an alternative of the above ML estimation based on the k-record values.
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