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Certain bivariate distributions and random processes connected with maxima and minima

Kozubowski, Tomasz J. and Podgórski, Krzysztof LU (2018) In Extremes
Abstract

The minimum and the maximum of t independent, identically distributed random variables have (Formula presented.) and Ft for their survival (minimum) and the distribution (maximum) functions, where (Formula presented.) and F are their common survival and distribution functions, respectively. We provide stochastic interpretation for these survival and distribution functions for the case when t > 0 is no longer an integer. A new bivariate model with these margins involve maxima and minima with a random number of terms. Our construction leads to a bivariate max-min process with t as its time argument. The second coordinate of the process resembles the well-known extremal process and shares with it the one-dimensional... (More)

The minimum and the maximum of t independent, identically distributed random variables have (Formula presented.) and Ft for their survival (minimum) and the distribution (maximum) functions, where (Formula presented.) and F are their common survival and distribution functions, respectively. We provide stochastic interpretation for these survival and distribution functions for the case when t > 0 is no longer an integer. A new bivariate model with these margins involve maxima and minima with a random number of terms. Our construction leads to a bivariate max-min process with t as its time argument. The second coordinate of the process resembles the well-known extremal process and shares with it the one-dimensional distribution given by Ft. However, it is shown that the two processes are different. Some fundamental properties of the max-min process are presented, including a distributional Markovian characterization of its jumps and their locations.

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author
organization
publishing date
type
Contribution to journal
publication status
epub
subject
keywords
Copula, Distribution theory, Exponentiated distribution, Extremal process, Extremes, Fréchet distribution, Generalized exponential distribution, Order statistics, Pareto distribution, Random maximum, Random minimum, Sibuya distribution
in
Extremes
pages
28 pages
publisher
Kluwer
external identifiers
  • scopus:85042111629
ISSN
1386-1999
DOI
10.1007/s10687-018-0311-2
language
English
LU publication?
yes
id
f1a5adcd-2f81-4471-bcb3-623587fbacce
date added to LUP
2018-03-06 08:31:34
date last changed
2018-05-29 09:51:48
@article{f1a5adcd-2f81-4471-bcb3-623587fbacce,
  abstract     = {<p>The minimum and the maximum of t independent, identically distributed random variables have (Formula presented.) and F<sup>t</sup> for their survival (minimum) and the distribution (maximum) functions, where (Formula presented.) and F are their common survival and distribution functions, respectively. We provide stochastic interpretation for these survival and distribution functions for the case when t &gt; 0 is no longer an integer. A new bivariate model with these margins involve maxima and minima with a random number of terms. Our construction leads to a bivariate max-min process with t as its time argument. The second coordinate of the process resembles the well-known extremal process and shares with it the one-dimensional distribution given by F<sup>t</sup>. However, it is shown that the two processes are different. Some fundamental properties of the max-min process are presented, including a distributional Markovian characterization of its jumps and their locations.</p>},
  author       = {Kozubowski, Tomasz J. and Podgórski, Krzysztof},
  issn         = {1386-1999},
  keyword      = {Copula,Distribution theory,Exponentiated distribution,Extremal process,Extremes,Fréchet distribution,Generalized exponential distribution,Order statistics,Pareto distribution,Random maximum,Random minimum,Sibuya distribution},
  language     = {eng},
  month        = {02},
  pages        = {28},
  publisher    = {Kluwer},
  series       = {Extremes},
  title        = {Certain bivariate distributions and random processes connected with maxima and minima},
  url          = {http://dx.doi.org/10.1007/s10687-018-0311-2},
  year         = {2018},
}