Hostname: page-component-84b7d79bbc-x5cpj Total loading time: 0 Render date: 2024-07-26T18:21:46.188Z Has data issue: false hasContentIssue false
  • English
  • Français

Max-infinite divisibility and multivariate total positivity

Part of: Distribution theory - Probability Multivariate analysis

Published online by Cambridge University Press:  14 July 2016

Abdulhamid A. Alzaid  and
Frank Proschan
Abdulhamid A. Alzaid*
Affiliation:
King Saud University
Frank Proschan*
Affiliation:
Florida State University
*
∗ Postal address: Department of Statistics and Operations Research, King Saud University, P.O. Box 2455, Riyadh-11451, Saudi Arabia.
∗∗ Postal address: Department of Statistics, Florida State University, Tallahassee, FL 32306-3033, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

The concept of max-infinite divisibility is viewed as a positive dependence concept. It is shown that every max-infinitely divisible distribution function is a multivariate totally positive function of order 2 (MTP2). Inequalities are derived, with emphasis on exchangeable distributions. Applications and examples are given throughout the paper.

Keywords

EXCHANGEABLE RANDOM VARIABLESMULTIVARIATE EXTREMAL PROCESSRELIABILITYSERIES SYSTEMPARALLEL SYSTEMEXTREME VALUE DISTRIBUTIONMULTIVARIATE DEPENDENCEINEQUALITIESMIN-INFINITE DIVISIBILITY

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 3 , September 1994 , pp. 721 - 730
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research partially supported by Grant AFOSR 91–0048.

Research supported by Grant AFOSR 91-0048.

References

Ahmed, A. N. and Alzaid, A. A. (1989) Weak association with stress-strength model application. Statist. Prob. Letters 8, 91–95.CrossRefGoogle Scholar
Balkema, A. A. and Resnick, S. I. (1977) Max-infinite divisibility. J. Appl. Prob. 14, 309–319.Google Scholar
Barlow, R. E. and Proschan, F. (1981) Statistical Theory of Reliability and Life Testing: Probability Models. To Begin With, Silver Spring, MD.Google Scholar
Block, H., Constigan, T. and Sampson, A. R. (1988) Product-type probability bounds of higher order. Technical Report No. 88 08, Dept. Math. Statist., University of Pittsburgh.Google Scholar
Block, H., Savits, T. and Shaked, M. (1982) Some concepts of negative dependence. Ann. Prob. 10, 765–772.Google Scholar
Ebrahimi, N. (1985) A stress-strength system. J. Appl. Prob. 22,467–472. Correction J. Appl. Prob. 24, 786.Google Scholar
Esary, J. D. and Proschan, F. (1972) Relationships among some notions of bivariate dependence. Ann. Math. Statist. 43, 651–655.Google Scholar
Glaz, J. and Johnson, B. Mck. (1984) Probability inequalities for multivariate distributions with dependence structures. J. Amer. Statist. Assoc. 79, 436–441.CrossRefGoogle Scholar
Glaz, J. and Johnson, B. Mck. (1986) Approximating boundary crossing probabilities with application to sequential tests. Sequential Analysis 1, 37–72.CrossRefGoogle Scholar
Harris, R. (1970) A multivariate definition of increasing hazard rate distribution functions. Ann. Math. Statist. 38, 713–717.Google Scholar
Karlin, S. (1968) Total Positivity. Stanford University Press.Google Scholar
Karlin, S. and Rinott, Y. (1980a) Classes of ordering measures and related correlation inequalities - I: multivariate totally positive distributions. J. Multivariate Anal. 10, 467–498.Google Scholar
Karlin, S. and Rinott, Y. (1980b) Classes of ordering measures and related correlation inequalities - II: multivariate reverse rule distribution. J. Multivariate Anal. 10, 499–516.Google Scholar
Marshall, A. and Olkin, I. (1967) A multivariate exponential distribution. J. Amer. Statist. Assoc. 62, 30–44.Google Scholar
Marshall, A. and Olkin, I. (1983) Domains of attraction of multivariate extreme value distributions. Ann. Prob. 11, 168–177.Google Scholar
Resnick, S. I. (1987) Extreme Values, Regular Variation, and Point Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
Shaked, M. (1977) A family of concepts of dependence for bivariate distribution. J. Amer. Statist. Assoc. 72, 642–650.Google Scholar
Tong, Y. L. (1980) Probability Inequalities in Multivariate Distributions. Academic Press, New York.Google Scholar