Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T05:26:41.631Z Has data issue: false hasContentIssue false
  • English
  • Français

Change-point models and conditionally pure birth processes: an inequality on the stochastic intensity

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

Published online by Cambridge University Press:  14 July 2016

Emilio De Santis  and
Fabio Spizzichino
Emilio De Santis*
Affiliation:
Università di Roma ‘La Sapienza’
Fabio Spizzichino*
Affiliation:
Università di Roma ‘La Sapienza’
*
∗ Postal address: Dipartimento di Matematica ‘Guido Castelnuovo’, Università di Roma ‘La Sapienza’, Piazzale Aldo Moro, 2, 00185 Roma, Italy
∗ Postal address: Dipartimento di Matematica ‘Guido Castelnuovo’, Università di Roma ‘La Sapienza’, Piazzale Aldo Moro, 2, 00185 Roma, Italy
Get access Rights & Permissions [Opens in a new window]

Abstract

We analyse several aspects of a class of simple counting processes that can emerge in some fields of applications where a change point occurs. In particular, under simple conditions we prove a significant inequality for the stochastic intensity.

Keywords

Change pointconditionally pure birth processrandom change of time-scaleload-sharing model

MSC classification

Primary: 60G55: Point processes 60J27: Continuous-time Markov processes on discrete state spaces 60K10: Applications (reliability, demand theory, etc.) 90B25: Reliability, availability, maintenance, inspection
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 939 - 952
Copyright
Copyright © Applied Probability Trust 2004 

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

Arjas, E., and Norros, I. (1991). Stochastic order and martingale dynamics in multivariate life length models: a review. In Stochastic Orders and Decision Under Risk (Hamburg, 1989; IMS Lecture Notes Monogr. Ser. 19), Institute of Mathematical Statistics, Hayward, CA, pp. 7–24.Google Scholar
Arjas, E., Haara, P., and Norros, I. (1992). Filtering the histories of a partially observed marked point process. Stoch. Process. Appl. 40, 225–250.Google Scholar
Aven, T., and Jensen, U. (1999). Stochastic Models in Reliability (Appl. Math. 41). Springer, New York.Google Scholar
Barbour, A. D., Holst, L., and Janson, S. (1992). Poisson Approximation (Oxford Studies Prob. 2). Oxford University Press.Google Scholar
Brémaud, P. (1981). Point Processes and Queues. Springer, New York.CrossRefGoogle Scholar
Carlstein, E., Müller, H.-G., and Siegmund, D. (eds) (1994). Change-Point Problems (AMS-IMS-SIAM Summer Res. Conf., Mount Holyoke College, South Hadley, MA, July 1992; IMS Lecture Notes Monogr. Ser. 23). Institute of Mathematical Statistics, Hayward, CA.Google Scholar
Herberts, T., and Jensen, U. (2004). Optimal detection of a change point in a Poisson process for different observation schemes. Scand. J. Statist. 31, 347–366.Google Scholar
Kurtz, T. G. (1982). Representation and approximation of counting processes. In Advances in Filtering and Optimal Stochastic Control (Lecture Notes Control Inf. Sci. 42), Springer, Berlin, pp. 177–191.CrossRefGoogle Scholar
Kwiecinski, A., and Szekli, R. (1996). Some monotonicity and dependence properties of self-exciting point processes. Ann. Appl. Prob. 6, 1211–1231.CrossRefGoogle Scholar
Muliere, P., and Scarsini, M. (1993). Some aspects of change-point problems. In Reliability and Decision Making (Siena, 1990), Chapman and Hall, London, pp. 273–285.Google Scholar
Müller, A., and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Ross, S. M. (1984). A model in which component failure rates depend only on the working set. Naval Res. Logistics Quart. 31, 297–300.CrossRefGoogle Scholar
Shaked, M., and Shanthikumar, J. G. (1991). Dynamic multivariate aging notions in reliability theory. Stoch. Process. Appl. 38, 85–97.Google Scholar
Shaked, M., Spizzichino, F., and Suter, F. (2002). Nonhomogeneous birth processes and l∞-spherical densities, with applications in reliability theory. Prob. Eng. Inf. Sci. 16, 271–288.Google Scholar
Spizzichino, F. (2001). Subjective Probability Models for Lifetimes (Monogr. Statist. Appl. Prob. 91). CRC, Boca Raton, FL.Google Scholar