Certain bivariate distributions and random processes connected with maxima and minima

Research output: Contribution to journalArticle

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 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.

Details

Authors
Organisations
External organisations
  • University of Nevada, Reno
Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Probability Theory and Statistics

Keywords

  • Copula, Distribution theory, Exponentiated distribution, Extremal process, Extremes, Fréchet distribution, Generalized exponential distribution, Order statistics, Pareto distribution, Random maximum, Random minimum, Sibuya distribution
Original languageEnglish
Pages (from-to)315-342
JournalExtremes
Volume21
Issue number2
Early online date2018 Feb 17
Publication statusPublished - 2018 Jun
Publication categoryResearch
Peer-reviewedYes

Related research output

Kozubowski, T. J. & Krzysztof Podgórski, 2016, Department of Statistics, Lund university, 24 p. (Working Papers in Statistics; no. 2016:9).

Research output: Working paper

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