Abstract
We study the class of renewal processes with Weibull lifetime distribution from the point of view of the general theory of point processes. We investigate whether a Weibull renewal process can be expressed as a Cox process. It is shown that a Weibull renewal process is a Cox process if and only if 0<α≤1, where α denotes the shape parameter of the Weibull distribution. The Cox character of the process is analyzed. It is shown that the directing measure of the process is continuous and singular.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berg, C. and Forst, G. (1975).Potential Theory on Locally Compact Abelian Groups, Springer, Berlin.
Feller, W. (1971).An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd ed., Wiley, New York.
Gnedenko, B. V. (1943). Sur la distribution limite du terme maximum d'une série aléatoire,Ann. of Math.,44, 423–453.
Grandell, J. (1976). Doubly stochastic Poisson processes,Lecture Notes in Math.,529, Springer, Berlin.
Grandell, J. (1991).Aspects of Risk Theory, Springer, New York.
Gumbel, E. J. (1958).Statistics of Extremes, Columbia University Press, New York.
Karr, A. F. (1991).Point Processes and Their Statistical Inference, 2nd ed., Marcel Dekker, New York.
Kingman, J. F. C. (1964). On doubly stochastic Poisson processes,Proceedings of the Cambridge Philosophical Society,60, 923–930.
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983).Extremes and Related Properties of Random Sequences and Processes, Springer, Berlin.
Mittnik, S. and Rachev, S. T. (1990). Alternative stable models for stock returns, Tech. Report, No. 131, Department of Statistics and Applied Probability, University of California, Santa Barbara.
Peto, R. and Lee, P. (1973). Weibull distributions for continuous-carcinogenesis experiments,Biometrics,29, 457–470.
Pike, M. C. (1966). A method of analysis of a certain class of experiments in carcinogenesis,Biometrics,22, 142–161.
Yannaros, N. (1988). On Cox processes and gamma renewal processes,J. Appl. Probab.,25, 423–427.
Yannaros, N. (1989). On Cox and renewal processes,Statist. Probab. Lett.,7.5, 431–433.
Author information
Authors and Affiliations
About this article
Cite this article
Yannaros, N. Weibull renewal processes. Ann Inst Stat Math 46, 641–648 (1994). https://doi.org/10.1007/BF00773473
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00773473