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An inaccuracy measure between non-explosive point processes with applications to Markov chains

Part of: Markov processes Stochastic processes Operations research and management science

Published online by Cambridge University Press:  25 October 2023

Vanderlei da Costa Bueno  and
Narayanaswamy Balakrishnan
Vanderlei da Costa Bueno*
Affiliation:
São Paulo University
Narayanaswamy Balakrishnan*
Affiliation:
McMaster University
*
*Postal address: Institute of Mathematics and Statistics, São Paulo University, Rua do Matão 1010, CEP 05508-090, São Paulo, Brazil. Email address: bueno@ime.usp.br
**Postal address: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4LS, Canada. Email address: bala@mcmaster.ca
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Abstract

Inaccuracy and information measures based on cumulative residual entropy are quite useful and have received considerable attention in many fields, such as statistics, probability, and reliability theory. In particular, many authors have studied cumulative residual inaccuracy between coherent systems based on system lifetimes. In a previous paper (Bueno and Balakrishnan, Prob. Eng. Inf. Sci. 36, 2022), we discussed a cumulative residual inaccuracy measure for coherent systems at component level, that is, based on the common, stochastically dependent component lifetimes observed under a non-homogeneous Poisson process. In this paper, using a point process martingale approach, we extend this concept to a cumulative residual inaccuracy measure between non-explosive point processes and then specialize the results to Markov occurrence times. If the processes satisfy the proportional risk hazard process property, then the measure determines the Markov chain uniquely. Several examples are presented, including birth-and-death processes and pure birth process, and then the results are applied to coherent systems at component level subject to Markov failure and repair processes.

Keywords

Cumulative residual inaccuracy measureMarkov chainsoccurrence timespoint processescoherent systems

MSC classification

Primary: 60G25: Prediction theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 90B25: Reliability, availability, maintenance, inspection
Type
Original Article
Information
Advances in Applied Probability , Volume 56 , Issue 2 , June 2024 , pp. 735 - 756
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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