Another characterization of the type I extreme value distribution
type: Item
instance of: scholarly article
title: Another characterization of the type I extreme value distribution
zbMATH Open document ID: 0616.62017
DOI: 10.1016/0167-7152(87)90060-5
author: Rocco Ballerini
published in: Statistics \& Probability Letters
publication date: 1987-01-01T00:00:00Z
review text: Let (1) \(Y_ n=X_ n+cn\), \(n=1,2,...\), \(c>0\), where \(\{X_ n\}\) is an i.i.d. sequence. In the special case when \(\{X_ n\}\) is i.i.d. with type I extreme value (or Gumbel) distribution \[ \Lambda_{\alpha,\beta}(x)=\exp \{-\exp \{-\beta^{-1}(x- \alpha)\}\},\quad -\infty <x<\infty, \] the author and \textit{S. I. Resnick} [Embedding sequences of successive maxima in extremal processes. J. Appl. Probab. (to appear)] showed that the maximum \(M_ n=\max \{Y_ 1,Y_ 2,...,Y_ n\}\) is independent of the indicator of a record at time n \(1_ n=1_{[Y_ n>M_{n-1}]}\) for each n. In this paper, the author proves that independence of \(M_ n\) and \(1_ n\) for models of the form (1) holds only when \(X_ i's\) have a Gumbel distribution.
reviewed by: Wiesław Dziubdziela
Mathematics Subject Classification ID: 62E10
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 3998964
zbMATH Keywords: characterization
zbMATH Keywords: type I extreme value distribution
zbMATH Keywords: record values
zbMATH Keywords: maximum
zbMATH Keywords: independence
zbMATH Keywords: Gumbel distribution
MaRDI profile type: Publication
full work available at URL: 0167-7152(87)90060-5
OpenAlex ID: W2029548190
cites work: Records from improving populations
cites work: Q5633348
cites work: Extremes and related properties of random sequences and processes
description: scientific article; zbMATH DE number 3998964