Abstract
The power law process (PLP) (i.e., the nonhomogeneous Poisson process with power intensity law) is perhaps the most widely used model for analyzing failure data from reliability growth studies. Statistical inferences and prediction analyses for the PLP with left-truncated data with classical methods were extensively studied by Yu et al. (2008) recently. However, the topics discussed in Yu et al. (2008) only included maximum likelihood estimates and confidence intervals for parameters of interest, hypothesis testing and goodness-of-fit test. In addition, the prediction limits of future failure times for failure-truncated case were also discussed. In this paper, with Bayesian method we consider seven totally different prediciton issues besides point estimates and prediction limits for xn+k. Specifically, we develop estimation and prediction methods for the PLP in the presence of left-truncated data by using the Bayesian method. Bayesian point and credible interval estimates for the parameters of interest are derived. We show how five single-sample and three two-sample issues are addressed by the proposed Bayesian method. Two real examples from an engine development program and a repairable system are used to illustrate the proposed methodologies.
Highlights
Crow (1974) extended the Duane model to the nonhomogeneous Poisson process (NHPP) with a power intensity law, which is known as AMSAA model due to its adoption by the U.S Army Materiel Systems Analysis Activity
If we adopt a noninformative prior distribution of λn (i.e., g(λn) ∝ 1/λn), the pseudo-posterior density of λn is f ∝ λnn∗/2−1 · exp the pseudo Bayes point estimator and the two-sided 100γ% Bayes credible interval (CI) for λn are respectively given by λn = n∗λM /a, (3.12)
Bayesian prediction limits of future failure times are available for both failure- and time-truncated cases
Summary
When failure times from different systems during their development programs are collected and analyzed, an approximate straight line pattern in the corresponding log-log plot of the cumulative mean time between failures (MTBF) against the cumulative operating time is usually observed (see, Duane, 1964). Crow (1974) extended the Duane model to the nonhomogeneous Poisson process (NHPP) with a power intensity law, which is known as AMSAA model due to its adoption by the U.S Army Materiel Systems Analysis Activity. Yu et al (2008) developed classical methods for statistical inferences and prediction analyses for the PLP with the first r − 1 failure times (i.e., {xi}ri=−11) being missing. This article aims to develop Bayesian estimation and prediction methods for the PLP with the first r − 1 failure times being missing. This left-truncated data pattern commonly occurs when (i) the importance of a reliability growth program is eventually recognized only when manufacturers reported the failures several times; and (ii) a new data-recording person may not be able to determine the exact failure times during the early stage of the process.
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