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Certain bivariate distributions and random processes connected with maxima and minima

Kozubowski, Tomasz J and Podgórski, Krzysztof LU (2016) In Working Papers in Statistics
Abstract
It is well-known that [S(x)]^n and [F(x)]^n are the survival function and the
distribution function of the minimum and the maximum of n independent, identically distributed random variables, where S and F are their common survival and distribution functions, respectively. These two extreme order statistics play important role in countless applications, and are the central and well-studied objects of extreme value theory. In this work we provide stochastic representations for the quantities [S(x)]^alpha and [F(x)]^beta, where alpha> 0 is no longer an integer, and construct a bivariate model with these margins. Our constructions and representations involve maxima and minima with a random number of terms. We also discuss... (More)
It is well-known that [S(x)]^n and [F(x)]^n are the survival function and the
distribution function of the minimum and the maximum of n independent, identically distributed random variables, where S and F are their common survival and distribution functions, respectively. These two extreme order statistics play important role in countless applications, and are the central and well-studied objects of extreme value theory. In this work we provide stochastic representations for the quantities [S(x)]^alpha and [F(x)]^beta, where alpha> 0 is no longer an integer, and construct a bivariate model with these margins. Our constructions and representations involve maxima and minima with a random number of terms. We also discuss generalizations to random process and further extensions. (Less)
Please use this url to cite or link to this publication:
author
and
organization
publishing date
type
Working paper/Preprint
publication status
published
subject
keywords
Copula, distribution theory, exponentiated distribution, extremes, generalized exponential distribution, order statistics, random minimum, random maximum, Sibuya distribution
in
Working Papers in Statistics
issue
2016:9
pages
24 pages
publisher
Department of Statistics, Lund university
language
English
LU publication?
yes
id
75d396f8-7067-474c-a25c-7f81bcdd3cd2
alternative location
http://journals.lub.lu.se/index.php/stat/article/view/16183
date added to LUP
2016-09-27 12:09:50
date last changed
2025-04-04 15:09:19
@misc{75d396f8-7067-474c-a25c-7f81bcdd3cd2,
  abstract     = {{It is well-known that [S(x)]^n and [F(x)]^n are the survival function and the<br>
distribution function of the minimum and the maximum of n independent, identically distributed random variables, where S and F are their common survival and distribution functions, respectively. These two extreme order statistics play important role in countless applications, and are the central and well-studied objects of extreme value theory. In this work we provide stochastic representations for the quantities [S(x)]^alpha and [F(x)]^beta, where  alpha&gt; 0 is no longer an integer, and construct a bivariate model with these margins. Our constructions and representations involve maxima and minima with a random number of terms. We also discuss generalizations to random process and further extensions.}},
  author       = {{Kozubowski, Tomasz J and Podgórski, Krzysztof}},
  keywords     = {{Copula; distribution theory; exponentiated distribution; extremes; generalized exponential distribution; order statistics; random minimum; random maximum; Sibuya distribution}},
  language     = {{eng}},
  note         = {{Working Paper}},
  number       = {{2016:9}},
  publisher    = {{Department of Statistics, Lund university}},
  series       = {{Working Papers in Statistics}},
  title        = {{Certain bivariate distributions and random processes connected with maxima and minima}},
  url          = {{https://lup.lub.lu.se/search/files/13092688/16183_41388_1_SM.pdf}},
  year         = {{2016}},
}