Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-06-01T19:00:36.686Z Has data issue: false hasContentIssue false
  • English
  • Français

Total Positivity of Markov Chains and the Failure Rate Character of Some First Passage Times

Part of: Distribution theory Markov processes Survival analysis and censored data

Published online by Cambridge University Press:  01 July 2016

Shiowjen Lee  and
J. Lynch
Shiowjen Lee*
Affiliation:
University of West Florida
J. Lynch*
Affiliation:
University of South Carolina
*
∗ Postal address; Department of Mathematics and Statistics, The University of West Florida, FL32514-5750, USA. Research partially supported by an NSF/EPSCOR grant.
∗∗ Postal address: Department of Statistics, University of South Carolina, Columbia, SC 29208, USA. Research partially supported by an NSF/EPSCOR grant and NSF grant no. DMS-9503104.
Get access Rights & Permissions [Opens in a new window]

Abstract

It is shown that totally positive order 2 (TP2) properties of the infinitesimal generator of a continuous-time Markov chain with totally ordered state space carry over to the chain's transition distribution function. For chains with such properties, failure rate characteristics of the first passage times are established. For Markov chains with partially ordered state space, it is shown that the first passage times have an IFR distribution under a multivariate total positivity condition on the transition function.

Keywords

FAILURE RATELOG-CONCAVITYTOTAL POSITIVITYMULTIVARIATE TOTAL POSITIVITY

MSC classification

Primary: 62E10: Characterization and structure theory
Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 62N05: Reliability and life testing
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 3 , September 1997 , pp. 713 - 732
Copyright
Copyright © Applied Probability Trust 1997 

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.)

References

Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Brown, M. and Chaganty, N. R. (1983) On the first passage time distribution for a class of Markov chains. Ann. Prob. 11, 1000–1008.Google Scholar
Çinlar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, New York.Google Scholar
Daniels, H. E. (1945) The statistical theory of the strength of bundles of threads I. Proc. R. Soc. A. 183, 405–435.Google Scholar
Drosen, J. W. (1986) Pure jump shock models in reliability. Adv. Appl. Prob. 18, 423–440.Google Scholar
Durham, S., Lynch, J. and Padgett, W. J. (1988) Inference for strength distributions of brittle fibers under increasing failure rate. J. Compos. Mat. 22, 1131–1140.Google Scholar
Durham, S., Lynch, J. and Padgett, W. J. (1990) TP2-orderings and the IFR property with applications. Prob. Eng. Inf. Sci. 4, 73–88.Google Scholar
Eaton, M. L. (1982) A review of selected topics in multivariate probability inequalities. Ann Statist. 10, 11–43.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986) Markov Processes-Characterization and Convergence. Wiley, New York.Google Scholar
Karlin, S. (1968) Total Positivity. Stanford University Press, Stanford, CA.Google Scholar
Karlin, S. and Rinott, Y. (1980) Classes of orderings of measures and related correlation inequalities. II. Multivariate reverse rule distributions. J. Multivar. Anal. 10, 467–498.CrossRefGoogle Scholar
Lynch, J., Mimmack, G. and Proschan, F. (1987) Uniform stochastic orderings and total positivity. Canadian J. Statist. 15, 63–69.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1988) On the first passage time of pure jump processes. J. Appl. Prob. 25, 501–509.Google Scholar
Shanthikumar, J. G. (1988) DFR property of first passage times and its preservation under geometric compounding. Ann Prob. 16, 397–406.Google Scholar