Title:

Three competing risk problems in the study of mechanical systems reliability

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 interarrival times of which are assumed to be independently distributed (but nonidentical)
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 subdistribution 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 shutdown is unknown for some
of the observations in the dataset.
In this thesis a new estimator based on the KaplanMeier product limit estimator is developed for
the above setup. A redistribution 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.
