This thesis considers three problems within the eld of competing risks modelling in reliability. The rst problem concerns the question of identi ability within certain subclasses of Doyen and Gaudoin's recently proposed generalised competing risks framework. Bedford and Lindqvist have shown identi ability for one such subclass - a two component series system in which, every time a component fails it is restored to a state "as good as new", while the other component is restored to a state "as bad as old". In this thesis two different subclasses are shown to be identifiable. The first is a generalisation of the Bedford and Lindqvist example for series systems with n components. The second is an n component series system in which each time a component fails it is restored to a state "as good as new". At the same time the remaining components are restored to a state "as good as new" with probability p (which may depend on both the component being restored and the component that failed), or to a state "as bad as old" with probability (1 - p). The second problem concerns the use of competing risks models to study opportunistic maintenance. Bedford and Alkali proposed the following model - the system exhibits a sequence of warning signals, the inter-arrival times of which are assumed to be independently distributed (but non-identical) exponential random variables. The hazard rate of the time to system failure is modelled as a piecewise exponential distribution, in which the hazard rate is constant between signals. A sequence of maintenance opportunities occurs according to a homogeneous Poisson process and the rst opportunity after the kth signal is used to preventatively maintain the system. In this thesis closed form expressions for the above model are calculated (subject to some minor technical restrictions) for the marginal distributions of the time to both preventative and corrective maintenance. Also, the sub-distribution of the time to corrective maintenance is calculated. The third problem concerns the estimation of the marginal distribution for one of two independent competing risks, when knowledge of which risk caused the system shut-down is unknown for some of the observations in the dataset. In this thesis a new estimator based on the Kaplan-Meier product limit estimator is developed for the above set-up. A re-distribution to the right algorithm is also developed and this is shown to be equivalent to the new estimator. The new estimator is also shown to be consistent.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:570538 |
| Date | January 2010 |
| Creators | Burnham, Michael Richard |
| Publisher | University of Strathclyde |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://oleg.lib.strath.ac.uk:80/R/?func=dbin-jump-full&object_id=16853 |
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