Certain bivariate distributions and random processes connected with maxima and minima
(2018) In Extremes 21(2). p.315-342- 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
- Kozubowski, Tomasz J. and Podgórski, Krzysztof LU
- organization
- publishing date
- 2018-06
- type
- Contribution to journal
- publication status
- published
- 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
- volume
- 21
- issue
- 2
- pages
- 315 - 342
- publisher
- Springer
- 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
- 2022-03-25 00:28:36
@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 > 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}}, keywords = {{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}}, number = {{2}}, pages = {{315--342}}, publisher = {{Springer}}, 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}}, doi = {{10.1007/s10687-018-0311-2}}, volume = {{21}}, year = {{2018}}, }