Krzysztof Podgórski
Professor, Head of the Department of Statistics
Certain bivariate distributions and random processes connected with maxima and minima
Author
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