Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-02-05T14:46:26.555Z Has data issue: false hasContentIssue false
  • English
  • Français

A NOTE ON THE CLASS OF GEOMETRIC COUNTING PROCESSES

Published online by Cambridge University Press:  28 March 2013

Ji Hwan Cha  and
Maxim Finkelstein
Ji Hwan Cha
Affiliation:
Department of Statistics, Ewha Womans University, Seoul, 120-750, Korea E-mail: [email protected]
Maxim Finkelstein
Affiliation:
Department of Mathematical Statistics, University of the Free State, 339 Bloemfontein 9300, South Africa E-mail: [email protected] and Max Planck Institute for Demographic Research, Rostock, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we suggest a new class of counting processes, called the Class of Geometric Counting Processes (CGCP), where each member of the counting process in the class has increments described by the geometric distribution. Distinct from the Poisson process, they do not possess the property of independent increments, which usually complicates probabilistic analysis. The suggested CGCP is defined and the dependence structure shared by the members of the class is discussed. As examples of useful applications, we consider stochastic survival models under external shocks. We show that the corresponding survival probabilities under reasonable assumptions can be effectively described by the CGCP without specifying the dependence structure.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 27 , Issue 2 , April 2013 , pp. 177 - 185
Copyright
Copyright © Cambridge University Press 2013

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

1.Cha, J.H. & Mi, J. (2007). Study of a stochastic failure model in a random environment. Journal of Applied Probability 44: 151–163.CrossRefGoogle Scholar
2.Cha, J.H. & Finkelstein, M. (2009). On a terminating shock process with independent wear increments. Journal of Applied Probability 46: 353–362.CrossRefGoogle Scholar
3.Cha, J.H. & Finkelstein, M. (2011). On new classes of extreme shock models and some generalizations. Journal of Applied Probability 48: 258–270.CrossRefGoogle Scholar
4.Ross, S.M. (1996). Stochastic processes. New York: John Wiley.Google Scholar
5.Beichelt, F. & Fischer, K. (1980). General failure model applied to preventive maintenance policies. IEEE Transactions on Reliability 29: 39–41.CrossRefGoogle Scholar
6.Block, H.W., Borges, W.S., & Savits, T.H. (1985). Age-dependent minimal repair. Journal of Applied Probability 22: 370–385.CrossRefGoogle Scholar
7.Finkelstein, M. (2003). Simple bounds for terminating Poisson and renewal processes. Journal of Statistical Planning and Inference 113: 541–548.CrossRefGoogle Scholar
8.Esary, J.D., Marshal, A.W., & Proschan, F. (1973). Shock models and wear processes. Annals of Probability 1: 627–649.CrossRefGoogle Scholar
9.Al-Hameed, M.S. & Proschan, F. (1973). Nonstationary shock models. Stochastic Processes and their Applications 1: 383–404.CrossRefGoogle Scholar
10.Nakagawa, T. (2007). Shock and damage models in reliability theory. London: Springer.Google Scholar
11.Finkelstein, M. (2008). Failure rate modelling for reliability and risk. London: Springer.Google Scholar