Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-24T05:15:57.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounds for the Joint Survival and Incidence Functions Through Coherent System Data

Part of: Survival analysis and censored data Distribution theory - Probability Operations research and management science

Published online by Cambridge University Press:  01 July 2016

J. V. Deshpande  and
S. R. Karia
J. V. Deshpande*
Affiliation:
University of Poona
S. R. Karia*
Affiliation:
University of Poona
*
∗ Postal Address: Department of Statistics, University of Poona, Pune-411007, India.
∗ Postal Address: Department of Statistics, University of Poona, Pune-411007, India.
Get access Rights & Permissions [Opens in a new window]

Abstract

In the series system (competing risks) set-up the observed data are generally accepted as the lifetime (T) and the identifier (δ) of the component causing the failure of the system. Peterson (1976) has provided bounds for the joint survival function of the component lifetimes in terms of the joint distribution of (T, δ). In the case of more complex coherent systems, there are various schemes of observation in the literature. In this paper we provide bounds for the joint and marginal survival functions of the component lifetimes in terms of the joint distribution of the data as obtained under existing and new schemes of observation. We also tackle the reverse problem of obtaining bounds for the joint distributions of the data for given marginal distributions of the component lifetimes and the distribution of the system lifetimes.

Keywords

AUTOPSY MODELCOHERENT SYSTEMSCONTINUOUS MONITORINGINCIDENCE FUNCTIONSMINIMAL PATH AND CUT SETSSUBSURVIVAL FUNCTION

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 90B25: Reliability, availability, maintenance, inspection 60E99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 2 , June 1997 , pp. 478 - 497
Copyright
Copyright © Applied Probability Trust 1997 

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

Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Doss, H., Freitag, S. and Proschan, F. (1989) Estimating jointly system and component reliability using a mutual censoring approach. Ann. Statist. 17, 764–782.Google Scholar
Meilijson, I. (1981) Estimation of the lifetime distribution of the parts from the autopsy statistics of the machine. J. Appl. Prob. 18, 829–838.Google Scholar
Nowik, S. (1990) Identifiability problems in coherent systems. J. Appl. Prob. 27, 862–872.Google Scholar
Peterson, A. V. (1976) Bounds for a joint distribution function with fixed sub distribution functions: applications to competing risks. Proc. Nat. Acad. Sci. 73, 1, 1–13.Google Scholar
Tsiatis, A. (1975) A non-identifiability aspect of the problem of competing risks. Proc. Nat. Acad. Sci. 72, 20–22.Google Scholar
Yakovlev, A. Yu. and Rachev, S. T. (1988) Bounds for crude survival probabilities within competing risks frame work and their statistical applications. Statist. Prob. Lett. 6, 389–394.Google Scholar