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Modelling of Marginally Regular Bivariate Counting Process and its Application to Shock Model

  • Ji Hwan Cha
  • Massimiliano Giorgio
Article
  • 17 Downloads

Abstract

In this paper, we develop a new class of bivariate counting processes that have ‘marginal regularity’ property. But, the ‘pooled processes’ in the developed class of bivariate counting processes are not regular. Therefore, the proposed class of processes allows simultaneous occurrences of two types of events, which can be applicable in practical modeling of counting events. Initially, some basic properties of the new class of bivariate counting processes will be discussed. Based on the obtained properties, the joint distributions of the numbers of events in time intervals will be derived and the dependence structure of the bivariate process will be discussed. Furthermore, the marginal and conditional processes will be studied. The application of the proposed bivariate counting process to a shock model will also be considered. In addition, the generalization to the multivariate counting processes will be discussed briefly.

Keywords

Reliability Complete intensity functions Dependence structure Marginal process Shock model 

Mathematics Subject Classification (2010)

Primary 60K10 Secondary 62P30 

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Notes

Acknowledgments

The authors would like to thank the Editor and referees for helpful comments and valuable suggestions, which have improved the presentation of this paper considerably. The authors acknowledge that one of the referees’ insightful advice has led Theorem 2 to a more general result on association. The work of the first author was supported by Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0093827). The work of the first author was also supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1A2B2014211).

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of StatisticsEwha Womans UniversitySeoulRepublic of Korea
  2. 2.Department of EngineeringUniversity of Campania Luigi VanvitelliAversaItaly

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