Abstract
The minimum and the maximum of t independent, identically distributed random variables have \(\bar F^{t}\) and Ft for their survival (minimum) and the distribution (maximum) functions, where \(\bar F = 1-F\) 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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Acknowledgments
The authors thank two referees for their remarks that fundamentally improved the original version. The research of the first author was partially funded by the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement no 318984 - RARE and the research of the second author was partially supported by Riksbankens Jubileumsfond Grant Dnr: P13-1024:1 and Swedish Research Council Grant Dnr: 2013-5180.
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Appendix
Appendix
In the following algorithm in R language random samples from the copula given in Proposition 3.8 are simulated. In it, first, it is decided whether a random value is from the singular or the absolutely continuous component. Then, in the continuous case, a random variate V is simulated from its marginal (3.19) and then a random variate U, conditionally on the obtained value V, is simulated from the conditional distribution (3.21).
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Kozubowski, T.J., Podgórski, K. Certain bivariate distributions and random processes connected with maxima and minima. Extremes 21, 315–342 (2018). https://doi.org/10.1007/s10687-018-0311-2
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DOI: https://doi.org/10.1007/s10687-018-0311-2
Keywords
- Copula
- Distribution theory
- Exponentiated distribution
- Extremal process
- Extremes
- Generalized exponential distribution
- Order statistics
- Random minimum
- Random maximum
- Sibuya distribution
- Pareto distribution
- Fréchet distribution