Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-18T15:24:15.050Z Has data issue: false hasContentIssue false
  • English
  • Français

Competing risks and independent minima: a marked point process approach

Published online by Cambridge University Press:  01 July 2016

Elja Arjas  and
Priscilla Greenwood
Elja Arjas*
Affiliation:
University of Oulu
Priscilla Greenwood*
Affiliation:
University of British Columbia
*
∗Department of Applied Mathematics and Statistics, University of Oulu, SF-90570 Oulu 57, Finland.
∗∗Department of Mathematics, University of British Columbia, Vancouver, B.C. Canada.
Get access Rights & Permissions [Opens in a new window]

Abstract

Given an indexed family of rate functions, we construct a family of independent random variables such that their minimum and the corresponding index form a marked point having the same rates. If the rates have common discontinuities, the random variables are conditionally independent, given a random allocation of these jumps. Between any two adjacent points of a marked point process there is a similar structure. Coherent functions in system lifetime theory provide examples.

Non-identifiability of multivariate lifetime distributions from mortality data is interpreted in this context.

Keywords

COMPETING RISKSINDEPENDENT MINIMAMARKED POINT PROCESSESMARTINGALE METHODSNON-IDENTIFIABILITY
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 4 , December 1981 , pp. 669 - 680
Copyright
Copyright © Applied Probability Trust 1981 

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

Aalen, O. (1978) Nonparametric inference for a family of counting processes. Ann. Statist. 6, 701–726.CrossRefGoogle Scholar
Arjas, E. (1981) The failure and hazard processes in multivariate reliability systems. Math. Operat. Res. To appear.Google Scholar
Brémaud, P. (1981) Point Processes and Queues: Martingale Dynamics. Springer-Verlag, Berlin.Google Scholar
Brémaud, P. and Jacod, J. (1977) Processus ponctuels et martingales: résultats récents sur la modélisation et le filtrage. Adv. Appl. Prob. 9, 362–416.Google Scholar
Esary, J. D. and Marshall, A. (1974) Multivariate distributions with exponential minimums. Ann. Statist. 2, 84–98.CrossRefGoogle Scholar
Evans, R. A. (1980) Independent or mutually exclusive. IEE Trans. Reliability 29, 289 (editorial).Google Scholar
Gill, R. D. (1980) Censoring and Stochastic Integrals. Mathematical Centre Tracts 124, Amsterdam.Google Scholar
Jacod, J. (1975) Multivariate point processes: Predictable projection, Radon–Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitsth. 31, 235–253.Google Scholar
Langberg, N., Proschan, F. and Quinzi, A. J. (1977) Converting dependent models into independent ones, with applications in reliability. In The Theory and Applications of Reliability, with Emphasis on Bayesian and Nonparametric Methods, 1, ed Shimi, I. and Tsokos, C. P., Academic Press, New York, 259–276.Google Scholar
Langberg, N., Proschan, F. and Quinzi, A. J. (1978) Converting dependent models into independent ones, preserving essential features. Ann. Prob. 6, 174–181.Google Scholar
Liptser, R. N. and Shiryayev, A. N. (1978) Statistics of Random Processes, II. Applications. Springer-Verlag, Berlin.Google Scholar
Meyer, P. A. (1972) Martingales and Stochastic Integrals. Lecture Notes in Mathematics 284, Springer-Verlag, Berlin.Google Scholar
Miller, D. R. (1977) A note on independence of multivariate lifetimes in competing risks models. Ann. Statist. 5, 576–579.CrossRefGoogle Scholar
Prentice, R. L., Kalbfleisch, J. D., Peterson, A. V. Jr., Flournoy, N., and Farewell, V. T. (1978) The analysis of failure times in presence of competing risks. Biometrics 34, 541–554.Google Scholar
Prentice, R. L. and Kalbfleisch, J. D. (1979) Hazard rate models with covariates. Biometrics 35, 25–39.CrossRefGoogle ScholarPubMed
Stroock, D. W. and Varadhan, S. R. S. (1979) Multidimensional Diffusion Processes. Springer-Verlag, New York.Google Scholar
Tsiatis, A. (1975) A nonidentifiability aspect in the problem of competing risks. Proc. Nat. Acad. Sci. USA 72, 20–22.Google Scholar