Fpa Frank Coolen

Bayesian reliability analysis with imprecise prior probabilities

The Bayesian framework for statistical inference offers the possibility of taking expert opinions into account, and is therefore attractive in practical problems concerning reliability of technical systems. Probability is the only language in which uncertainty can be consistently expressed, and this requires the use of prior distributions for reporting expert opinions. In this paper an extension of the standard Bayesian approach based on the theory of imprecise probabilities and intervals of measures is developed. It is shown that this is necessary to take the nature of experts knowledge into account. The application of this approach in reliability theory is outlined. The concept of imprecise probabilities allows us to accept a range of possible probabilities from an expert for events of interest and thus makes the elicitation of prior information simpler and clearer. The method also provides a consistent way for combining the opinions of several experts.

Nonparametric predictive inference in statistical process control

Abstract Statistical process control (SPC) is used to decide when to stop a process as confidence in the quality of the next item(s) is low. Information to specify a parametric model is not always available, and as SPC is of a predictive nature, we present a control chart developed using nonparametric predictive inference. The proposed ‘extrema chart’, based on the extrema of a sample of observations from the process, is a generalisation of an existing nonparametric method, which controls a process using single observations. We examine the average run length (ARL) of both the one-sided and two-sided extrema chart, and a simulation study is presented to compare the extrema chart with the well known X¯ chart and CUSUM chart. The disadvantage of these charts is that when the process mean and variation of the in-control process have to be estimated, the ARL is biased. This is not an issue for the extrema chart, as no knowledge about the underlying distribution is required.

Imprecise conjugate prior densities for the one-parameter exponential family of distributions

Reconsidering generalizations of the original Bayesian framework that have been suggested during the last three decades, imprecise conjugate prior densities are proposed for members of the one-parameter exponential family of distributions.

A Bayes-competing risk model for the use of expert judgment in reliability estimation

In an analysis of the reliability of heat exchangers, a number of causes of failure and a number of potential failure patterns were identified. To allow a simplification of the analysis, five independent groups of failure causes were defined and formed the basis of a competing risks model. Further, lack of data and poor quality data required the experience of ‘experts’ to be used in a quantitative way through Bayesian methods. Lastly, to simplify the elicitation of the prior densities, a shape, scale and location parameter model for the failure time distributions was adopted.

Bayesian estimation of location parameters in life distributions

For some distributions commonly used in reliability analysis the likelihood function has a singularity at the smallest observation. The use of grouped likelihood in Bayesian inference is discussed. It is shown that the product of spacings can be used in place of the likelihood. Similarities with and differences from a standard Bayesian approach are illustrated by means of examples. In particular some computational advantages are pointed out.