Parametric and Semiparametric Models with Applications to Reliability, Survival Analysis, and Quality of Life pp 473-486 | Cite as
On Parameter Estimation for a Position-Dependent Marking of a Doubly Stochastic Poisson Process
Chapter
Abstract
For analyzing reliability of technical systems it is often important to investigate damage processes. In this paper we describe a damage process (Zt) which is assumed to be generated by a positionependent marking of a doubly stochastic Poisson process. For some parametric intensity kernels of the corresponding marked point process we determine maximum-likelihood estimations. Censored observations are taken into account. Furthermore, the large sample case is considered.1
Keywords
Marked point process doubly stochastic Poisson process position-dependent marking shock model parametric models maximum-likelihood estimation asymptotic propertiesPreview
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References
- 1.Albrecht, P. (1981). Dynamische Statistische Entscheidungsverfahren für Schadenzahlprozesse, VVW Karlsruhe, Germany.Google Scholar
- 2.Anderson, P., Borgan, ø., Gill, R. and Keiding, N. (1993). Statistical Models Based on Counting Processes, Springer-Verlag, New York.CrossRefGoogle Scholar
- 3.Aven, T. and Jensen, U. (1998). Stochastic Models in Reliability, Springer-Verlag, New York.Google Scholar
- 4.Bremaud, P. (1981). Point Processes and Queues, Springer-Verlag, New York.MATHCrossRefGoogle Scholar
- 5.Cox, D. R. (1955). Some statistical methods connected with series of events. Journal of the Royal Statistical Society, Series B, 17, 129–164.MATHGoogle Scholar
- 6.Cramer, H. (1969). On streams of random events. Skandinavisk Aktuarietidskrift Suppl., 85, 13–23.MathSciNetGoogle Scholar
- 7.Esary, J. D., Marshall, A. W. and Proshan, F. (1973). Shock models and wear processes, Annals of Probability, 1, 627–649.MATHCrossRefGoogle Scholar
- 8.Feng, W., Adachi, K. and Kowada, M. (1994). Optimal replacement under additive damage in a Poisson random environment, Communications in Statistics — Stochastic Models, 10, 679–700.MATHMathSciNetCrossRefGoogle Scholar
- 9.Grandell, J. (1991). Aspects of Risk Theory, Springer-Verlag, New York.MATHCrossRefGoogle Scholar
- 10.Grandell, J. (1997). Mixed Poisson Processes, Chapman & Hall, London, England.MATHCrossRefGoogle Scholar
- 11.König, D. and Schmidt, V. (1992). Zufällige Punktprozesse, B.G. Teubner, Stuttgart, Germany.MATHCrossRefGoogle Scholar
- 12.Last, G. and Brandt, A. (1995). Marked Point Processes on the Real Line, Springer-Verlag, New York.MATHGoogle Scholar
- 13.Pieper, V. and Tiedge, J. (1983). Zuverlässigkeitsmodelle auf der Grundlage stochastischer Modelle von Verschleißprozessen, Mathenatuscg Operationsforschung und Statistik, Series Statistics, 14, 485–502.MATHMathSciNetGoogle Scholar
- 14.Schröter, K.J. (1995). Verfahren zur Approximation der Gesamtschadenverteilung, VVW Karlsruhe, Germany.Google Scholar
- 15.Shaked, M. (1983). Wear and damage processes from shock models, In Proceedings of the Symposium on Reliability Theory and Models, pp. 43–64, Charlotte, North Carolina.Google Scholar
- 16.Sobczyk, K. (1987). Stochastic models for fatigue damage of materials. Advances in Applied Probability, 19, 652–673.MATHMathSciNetCrossRefGoogle Scholar
- 17.Wendt, H. (1999). Parameterschätzungen für eine Klasse doppelt-stochastischer Poisson Prozesse bei unterschiedlichen Beobachtungsinformationen, Ph.D. Thesis.Google Scholar
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