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Portrait of Krzysztof Podgórski. Photo.

Krzysztof Podgórski

Professor, Head of the Department of Statistics

Portrait of Krzysztof Podgórski. Photo.

Certain bivariate distributions and random processes connected with maxima and minima

Author

  • Tomasz J. Kozubowski
  • Krzysztof Podgórski

Summary, in English

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.

Department/s

  • Department of Statistics

Publishing year

2018-06

Language

English

Pages

315-342

Publication/Series

Extremes

Volume

21

Issue

2

Document type

Journal article

Publisher

Springer

Topic

  • 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

Status

Published

ISBN/ISSN/Other

  • ISSN: 1386-1999