Hostname: page-component-77c78cf97d-tlp4c Total loading time: 0 Render date: 2026-04-24T17:47:33.833Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A MULTIVARIATE GENERALIZED POLYA PROCESS WITHOUT REGULARITY PROPERTY

Published online by Cambridge University Press:  24 April 2019

Ji Hwan Cha  and
F.G. Badía
Ji Hwan Cha
Affiliation:
Department of Statistics, Ewha Womans University, Seoul 120-750, Republic of Korea E-mail: jhcha@ewha.ac.kr
F.G. Badía
Affiliation:
Department of Statistical Methods, University of Zaragoza, Zaragoza 50018, Spain E-mail: gbadia@unizar.es
Get access Rights & Permissions [Opens in a new window]

Abstract

Most of the multivariate counting processes studied in the literature are regular processes, which implies, ignoring the types of the events, the non-occurrence of multiple events. However, in practice, several different types of events may occur simultaneously. In this paper, a new class of multivariate counting processes which allow simultaneous occurrences of multiple types of events is suggested and its stochastic properties are studied. For the modeling of such kind of process, we rely on the tool of superposition of seed counting processes. It will be shown that the stochastic properties of the proposed class of multivariate counting processes are explicitly expressed. Furthermore, the marginal processes are also explicitly obtained. We analyze the multivariate dependence structure of the proposed class of counting processes.

Keywords

characterization of multivariate counting processescomplete intensity functionsdependence structuregeneralized polya processsuperposition

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Issue 4 , October 2020 , pp. 484 - 506
Copyright
Copyright © Cambridge University Press 2019

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.)

Article purchase

Temporarily unavailable

References

1.Aven, T. & Jensen, U. (1999). Stochastic models in reliability. New York: Springer.CrossRefGoogle Scholar
2.Aven, T. & Jensen, U. (2000). A general minimal repair model. Journal of Applied Probability 37: 187–197.CrossRefGoogle Scholar
3.Barbu, V.S. & Limnios, N. (2008). Semi-Markov chains and hidden semi-Markov models toward applications (their use in reliability and DNA analysis). New York: SpringerGoogle Scholar
4.Buhlmann, H. (1970). Mathematical methods in risk theory. Berlin: Springer.Google Scholar
5.Cai, J. & Li, H. (2005). Multivariate risk model of phase type. Insurance: Mathematics and Economics 36: 137–152.Google Scholar
6.Campbell, J.T. (1934). The Poisson correlation function. Proceedings of the Edinburgh Mathematical Society, Series 2 4: 18–26.10.1017/S0013091500024135CrossRefGoogle Scholar
7.Cha, J.H. (2014). Characterization of the generalized Polya process and its applications. Advances in Applied Probability 46: 1148–1171.10.1239/aap/1418396247CrossRefGoogle Scholar
8.Cha, J.H. (2015). Variables acceptance reliability sampling plan for repairable items. Statistics 49: 1141–1156.CrossRefGoogle Scholar
9.Cha, J.H. & Finkelstein, M. (2016). On some mortality rate processes and mortality deceleration with age. Journal of Mathematical Biology 72: 331–342.10.1007/s00285-015-0885-0CrossRefGoogle ScholarPubMed
10.Cha, J.H. & Finkelstein, M. (2018). Point processes for reliability analysis (shocks and repairable systems). London: Springer.10.1007/978-3-319-73540-5CrossRefGoogle Scholar
11.Cha, J.H. & Giorgio, M. (2016). On a class of multivariate counting processes. Advances in Applied Probability 48: 443–462.10.1017/apr.2016.9CrossRefGoogle Scholar
12.Chan, W.S., Hailiang Yang, H.Zhang, L. (2003). Some results on ruin probabilities in a two-dimensional risk model. Insurance: Mathematics and Economics 32: 345–358.Google Scholar
13.Çinlar, E. (1975). Introduction to Stochastic processes. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
14.Cox, D.R. & Lewis, P.A.W. (1972). Multivariate point processes. in Proceedings of the Sixth Berkeley Symposium in Mathematical Statistics (LeCam, L.M. Ed.), pp. 401–448.Google Scholar
15.Cuadras, C.M. (2002). On the covariance between functions. Journal of Multivariate Analysis 81: 19–27.10.1006/jmva.2001.2000CrossRefGoogle Scholar
16.Holgate, B. (1964). Estimation for the bivariate Poisson distribution. Biometrika 51: 241–245.CrossRefGoogle Scholar
17.Joe, H. (1997). Multivariate models and dependence concepts. London: Chapman and Hall.CrossRefGoogle Scholar
18.Kundu, D. & Gupta, R.D. (2009). Bivariate generalized exponential distribution. Journal of Multivariate Analysis 100: 581–593.CrossRefGoogle Scholar
19.Kundu, D. & Gupta, R.D. (2010). Modified Sarhan–Balakrishnan singular bivariate distribution. Journal of Statistical Planning and Inference 140: 526–538.10.1016/j.jspi.2009.07.026CrossRefGoogle Scholar
20.Limnios, N. & Oprişan, G. (2001). Semi-Markov processes and reliability. Boston: Birkhäuser.CrossRefGoogle Scholar
21.Marshall, A.W. & Olkin, I. (1967). A multivariate exponential distribution. Journal of American Statistical Association 62: 30–44.CrossRefGoogle Scholar
22.Müller, A. & Stoyan, D. (2002). Comparison methods for Stochastic models and risks. Chichester: Wiley.Google Scholar
23.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. New York: Springer.CrossRefGoogle Scholar
24.Willmot, G.E. & Woo, J.K. (2007). On the class of Erlang mixtures with risk theoretic applications. North American Actuarial Journal 11: 99–115.CrossRefGoogle Scholar
25.Willmot, G.E. & Woo, J.K. (2015). On some properties of a class of multivariate Erlang mixtures with insurance applications. ASTIN Bulletin: The Journal of the IAA 45: 151–173.CrossRefGoogle Scholar
26.Woo, J.K. (2016). On multivariate discounted compound renewal sums with time-dependent claims in the presence of reporting/payment delays. Insurance: Mathematics and Economics 70: 354–363.Google Scholar