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Gamma lifetimes and one-shot device testing analysis

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  • Balakrishnan, N.
  • Ling, M.H.

Abstract

Gamma distribution is widely used to model lifetime data in reliability and survival analysis. In the context of one-shot device testing, encountered commonly in testing devices such as munitions, rockets, and automobile air-bags, either left- or right-censored data are collected instead of actual lifetimes of the devices under test. The destructive nature of one-shot devices makes it difficult to collect sufficient lifetime information on the devices. For this reason, accelerated life-tests are commonly used in which the test devices are subjected to conditions in excess of its normal use-condition in order to induce more failures, so as to obtain more lifetime information within a relatively short period of time. In this paper, we discuss the analysis of one-shot device testing data under accelerated life-tests based on gamma distribution. Both scale and shape parameters of the gamma distribution are related to stress factors through log–linear link functions. Since lifetimes of devices under this test are censored, the EM algorithm is developed here for the estimation of the model parameters. The inference on the reliability at a specific mission time as well as on the mean lifetime of the devices is also developed. Moreover, by using missing information principle, the asymptotic variance–covariance matrix of the maximum likelihood estimates under the EM framework is determined, and is then used to construct asymptotic confidence intervals for the parameters of interest. For the reliability at a specific mission time and also for the mean lifetime of the devices, transformation approaches are proposed for the construction of confidence intervals. These confidence intervals are then compared through a simulation study in terms of coverage probabilities and average widths. Recommendations are then made for an appropriate approach for the construction of confidence intervals for different sample sizes and different levels of reliability. A distance-based statistic is suggested for testing the validity of the model to an observed data. Finally, since current status data with covariates in survival analysis and one-shot device testing data with stress factors in reliability analysis share the same data structure, a real data from a toxicological study is used to illustrate the developed methods.

Suggested Citation

  • Balakrishnan, N. & Ling, M.H., 2014. "Gamma lifetimes and one-shot device testing analysis," Reliability Engineering and System Safety, Elsevier, vol. 126(C), pages 54-64.
  • Handle: RePEc:eee:reensy:v:126:y:2014:i:c:p:54-64
    DOI: 10.1016/j.ress.2014.01.009
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    References listed on IDEAS

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    1. Dianne M. Finkelstein & Louise M. Ryan, 1987. "Estimating Carcinogenic Potency from a Rodent Tumorigenicity Experiment," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 36(2), pages 121-133, June.
    2. Balakrishnan, N. & Ling, M.H., 2012. "EM algorithm for one-shot device testing under the exponential distribution," Computational Statistics & Data Analysis, Elsevier, vol. 56(3), pages 502-509.
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    Cited by:

    1. Ranjan, Rakesh & Sen, Rijji & Upadhyay, Satyanshu K., 2021. "Bayes analysis of some important lifetime models using MCMC based approaches when the observations are left truncated and right censored," Reliability Engineering and System Safety, Elsevier, vol. 214(C).
    2. Franko, Mitja & Nagode, Marko, 2015. "Probability density function of the equivalent stress amplitude using statistical transformation," Reliability Engineering and System Safety, Elsevier, vol. 134(C), pages 118-125.
    3. N. Balakrishnan & E. Castilla & N. Martín & L. Pardo, 2019. "Robust estimators for one-shot device testing data under gamma lifetime model with an application to a tumor toxicological data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(8), pages 991-1019, November.
    4. Zhu, Xiaojun & Liu, Kai & He, Mu & Balakrishnan, N., 2021. "Reliability estimation for one-shot devices under cyclic accelerated life-testing," Reliability Engineering and System Safety, Elsevier, vol. 212(C).
    5. Wu, Shuo-Jye & Hsu, Chu-Chun & Huang, Syuan-Rong, 2020. "Optimal designs and reliability sampling plans for one-shot devices with cost considerations," Reliability Engineering and System Safety, Elsevier, vol. 197(C).
    6. Leijia Ding & Wenhao Gui, 2023. "Statistical Inference of Two Gamma Distributions under the Joint Type-II Censoring Scheme," Mathematics, MDPI, vol. 11(9), pages 1-23, April.
    7. Ling, M.H. & Hu, X.W., 2020. "Optimal design of simple step-stress accelerated life tests for one-shot devices under Weibull distributions," Reliability Engineering and System Safety, Elsevier, vol. 193(C).
    8. Qin, Shuidan & Wang, Bing Xing & Wu, Wenhui & Ma, Chao, 2022. "The prediction intervals of remaining useful life based on constant stress accelerated life test data," European Journal of Operational Research, Elsevier, vol. 301(2), pages 747-755.
    9. Balakrishnan, N. & So, H.Y. & Ling, M.H., 2015. "EM algorithm for one-shot device testing with competing risks under exponential distribution," Reliability Engineering and System Safety, Elsevier, vol. 137(C), pages 129-140.
    10. Man-Ho Ling & Narayanaswamy Balakrishnan & Chenxi Yu & Hon Yiu So, 2021. "Inference for One-Shot Devices with Dependent k -Out-of- M Structured Components under Gamma Frailty," Mathematics, MDPI, vol. 9(23), pages 1-24, November.
    11. Zhu, Xiaojun & Balakrishnan, N., 2022. "One-shot device test data analysis using non-parametric and semi-parametric inferential methods and applications," Reliability Engineering and System Safety, Elsevier, vol. 221(C).

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