Prediction Intervals for Reliability Growth Models with Small Sample Sizes

Authors: Quigley John, Walls Lesley

Management Science Working Paper No. 13 (2004)

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Abstract

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Reliability growth testing is conducted during product development to flush out weaknesses and hence mature the system design. There exists a dilemma in that better reliability means fewer observed failures and small samples imply limited information upon which to base development decisions. In particular, basing decisions on point estimates of reliability is unsatisfactory because it fails to account for uncertainty in the model used to describe the failure time data and the process generating the failure time data. Typically, probability models are used to describe the variability of the process, while interval estimates are used to describe the knowledge uncertainty in model. In this chapter we combine these two sources of uncertainty to develop methods for generating prediction intervals for use in reliability growth analysis. The approach to modelling growth uses a hybrid of Bayesian and Classical approaches to statistical inference. A prior distribution is used to describe the number of potential faults believed to exist within a system design, while reliability growth test data is used to estimate the rate at which these faults are detected. An industry standard model is described and extended to account for both forms of uncertainty in supporting predictions about the time to the detection of the next fault. After deriving the prediction intervals, an analysis of the statistical properties of the underlying distribution is presented for a range of small sample sizes. An illustrative example is given to demonstrate the computation and interpretation of the prediction intervals within a typical product development process.