Statistics Seminars: Survival Models and Threshold Crossings
7 August 2009 11:00 in CM107
Many reliability models are based on threshold crossings. The gamma process and extreme value theory are key tools in these models. In this talk I review some of the aspects of the models and show that there are generalizations with wider application.
I examine the formulation of the models in terms of system state and models for time to failure. There is the natural and well known duality between these models, but it is also possible to generalize and relax assumptions. The models are usually formulated in terms of a stochastic process. The events of interest are defined by the entry or exit of the process into or out of a critical set.
State space models are usually built from variations on shock models with additive increments and an assumption monotonicity. Time domain models are then derived by examining, for example, determining first hitting times. By dropping the requirement of monotonicity the scope of the models can be widened; the construction can be achieved by suppressing the state space and working from arbitrary survival distributions. Some degradation models also give rise naturally to processes with multiplicative increments and can be described by, for example, geometric Brownian motions. The multiplicative models are members of a family of growth models.
Lastly, the evolution of multiple degradation processes is considered. When the initiation of the processes is a Poisson process it is shown that elementary arguments using the classical colouring theorem for Poisson processes captures all aspects of interest for such a model.
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