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Hazard rate and reversed hazard rate monotonicities in continuous-time Markov chains

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Masaaki Kijima
Masaaki Kijima*
Affiliation:
Tokyo Metropolitan University
*
∗Postal address: Faculty of Economics, Tokyo Metropolitan University, 1–1 Minami-Ohsawa, Hachiohji, Tokyo 192–0397, Japan. Email address: kijima@bcomp.metro-u.ac.jp.
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Abstract

A continuous-time Markov chain on the non-negative integers is called skip-free to the right (left) if only unit increments to the right (left) are permitted. If a Markov chain is skip-free both to the right and to the left, it is called a birth–death process. Karlin and McGregor (1959) showed that if a continuous-time Markov chain is monotone in the sense of likelihood ratio ordering then it must be an (extended) birth–death process. This paper proves that if an irreducible Markov chain in continuous time is monotone in the sense of hazard rate (reversed hazard rate) ordering then it must be skip-free to the right (left). A birth–death process is then characterized as a continuous-time Markov chain that is monotone in the sense of both hazard rate and reversed hazard rate orderings. As an application, the first-passage-time distributions of such Markov chains are also studied.

Keywords

Monotone Markov chainbirth–death processstochastic orderingfirst-passage-time distribution

MSC classification

Primary: 60J35: Transition functions, generators and resolvents
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 3 , September 1998 , pp. 545 - 556
Copyright
Copyright © Applied Probability Trust 1998 

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