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Lifetime distribution selection for complete and censored multi-level testing data and its influence on probability of failure estimates

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Abstract

A general method of lifetime distribution selection for multi-level complete and censored testing data is developed. To alleviate the scarcity of test samples at each individual stress levels, a mathematical model is incorporated to correlate the lifetimes and stresses allowing for utilizing the multi-level lifetime data as a whole. Probabilistic modeling of the model prediction error is made and the equivalence between the lifetime distribution and the error distribution is shown. Based on the error modeling and likelihood functions, a Bayesian framework for lifetime distribution selection is proposed. It is shown that the classical information criterion is a local measure and the Bayes factor is a global measure over the likelihood space. A two-step assessment procedure integrating the goodness-of-fit and asymptotic Bayes factors is given. The proposed method is demonstrated using three engineering examples. Comparisons between the proposed method and the conventional methods such as ratio test and the difference of information criteria are made. The influence of the distribution selection on the subsequent probability of failure estimates is discussed.

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Acknowledgments

The support is greatly acknowledged. The authors would like to thank the anonymous reviewers for their constructive comments.

Funding

The work in this study was supported by the Science Challenge Project, No.TZ2018007, and by National Natural Science Foundation of China, Nos. 51975546, 11872088.

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Authors and Affiliations

  1. School of Reliability and Systems Engineering, Beihang University, 37 Xueyuan Rd., Beijing, 100191, China

    Jingjing He

  2. Graduate School of China Academy of Engineering Physics, 10 Xibeiwang E. Rd., Beijing, 100193, China

    Jie Chen & Xuefei Guan

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  1. Jingjing He
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  2. Jie Chen
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  3. Xuefei Guan
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Corresponding author

Correspondence to Xuefei Guan.

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Replication of results

The calculations in this study are made using a MATLAB script-compatible software package GNU Octave version 5.1.0.

  1. 1.

    Asymptotic Bayes factors

    In the calculation of asymptotic Bayes factors, the modes of the logarithm of a distribution is found using function fminsearch1 with a relative error tolerance 10− 8 and an initial value obtained using the least square method. The censored data are treated as complete data in the least square method only for the purpose of getting an approximate initial value for fminsearch. Once the modes are found, the Hessian matrices evaluated at the modes are obtained using a numerical differentiation script suite called DERIVEST2 provided by John R. D’Errico, which runs directly in GNU Octave.

    1https://octave.sourceforge.io/octave/function/fminsearch.html

    2https://www.convexoptimization.com/TOOLS/DERIVEST.pdf

  2. 2.

    Confidence intervals and histograms

    The calculation of confidence intervals of stress-life model predictions and σ histograms are made using MCMC samples drawn using the standard Metropolis-Hastings (MH) algorithm. The starting sample in the MH is set as the initial value used in fminsearch. The proposal density of the MH is the Gaussian distribution with a zero mean and a standard deviation set as half of the estimated standard deviation by the least square method mentioned above. A total of 5 × 106 samples are obtained for each cases.

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He, J., Chen, J. & Guan, X. Lifetime distribution selection for complete and censored multi-level testing data and its influence on probability of failure estimates. Struct Multidisc Optim 62, 1–17 (2020). https://doi.org/10.1007/s00158-019-02465-7

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  • DOI: https://doi.org/10.1007/s00158-019-02465-7

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