Average Life Prediction Based on Incomplete Data

ABSTRACT

The two-parameter exponential distribution can often be used to describe the lifetime of products for example, electronic components, engines and so on. This paper considers a prediction problem arising in the life test of key parts in high speed trains. Employing the Bayes method, a joint prior is used to describe the variability of the parameters but the form of the prior is not specified and only several moment conditions are assumed. Under the condition that the observed samples are randomly right censored, we define a statistic to predict a set of future samples which describes the average life of the second-round samples, firstly, under the condition that the censoring distribution is known and secondly, that it is unknown. For several different priors and life data sets, we demonstrate the coverage frequencies of the proposed prediction intervals as the sample size of the observed and the censoring proportion change. The numerical results show that the prediction intervals are efficient and applicable.

The two-parameter exponential distribution can often be used to describe the lifetime of products for example, electronic components, engines and so on. This paper considers a prediction problem arising in the life test of key parts in high speed trains. Employing the Bayes method, a joint prior is used to describe the variability of the parameters but the form of the prior is not specified and only several moment conditions are assumed. Under the condition that the observed samples are randomly right censored, we define a statistic to predict a set of future samples which describes the average life of the second-round samples, firstly, under the condition that the censoring distribution is known and secondly, that it is unknown. For several different priors and life data sets, we demonstrate the coverage frequencies of the proposed prediction intervals as the sample size of the observed and the censoring proportion change. The numerical results show that the prediction intervals are efficient and applicable.

Cite this paper

nullT. Tang, L. Wang, F. Wu and L. Wang, "Average Life Prediction Based on Incomplete Data,"*Applied Mathematics*, Vol. 2 No. 1, 2011, pp. 93-105. doi: 10.4236/am.2011.21011.

nullT. Tang, L. Wang, F. Wu and L. Wang, "Average Life Prediction Based on Incomplete Data,"

References

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[4] S. Geisser, “Predictive Inference: An Introduction,” Chapman and Hall, London, 1993.

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[8] W. T. Huang and H. H. Huang, “Empirical Bayes Estimation of the Guarantee Lifetime in a Two-Parameter Exponential Distribution,” Statistics and Probability Letters, Vol. 76, No. 16, 2006, pp. 1821-1829. doi:10.1016/ j.spl.2006.04.034

[9] J. W. Wu, H. M. Lee and C. L. Lei, “Computational Testing Algorithmic Procedure of Assessment for Lifetime Performance Index of Products with Two-Parameter Exponential Distribution,” Applied Mathematics and Computation, Vol. 190, No. 5, 2007, pp. 116-125. doi: 10.1016/j.amc.2007.01.010

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[11] R. S. Singh, “Bayes and Empirical Bayes Procedures for Selecting Good Populations from a Translated Exponential Family,” Empirical Bayes and Likelihood Inference, Springer, New York, 2001.

[12] H. Lian, “Consistency of Bayesian Estimation of a Step Function,” Statistics and Probability Letters, Vol. 77, No. 1, 2007, pp. 19-24. doi:10.1016/j.spl.2006.05.007

[13] Q. H. Wang, “Empirical Likelihood for a Class of Functions of Survival Distribution with Censored Data,” Annals of the Institute of Statistical Mathematics, Vol. 53, No. 3, 2001, pp. 517-527. doi:10.1023/A:1014617112870

[14] E. T. Kaplan and P. Meier, “Nonparametric Estimation from Incomplete Observations,” Journal of the American Statistical Association, Vol. 53, No. 282, 1958, pp. 457-481. doi:10.2307/2281868

[15] R. Y. Zheng, “Improvement on Grey Modelling Approach and Its Applications to Fatigue Reliability Research,” Master Dissertation, National University of Defense Technology, Chang Sha, 1989.

[16] K. O. Bowman and L. R. Shenton, “Weibull Distributions When the Shape Parameter is Defined,” Computational Statistics and Data Analysis, Vol. 36, No. 4, 2001, pp. 299-310. doi:10.1016/S0167-9473(00)00048-7

[17] U. Singh, P. K. Gupta and S. K. Upadhyay, “Estimation of Parameters for Exponentiated-Weibull Family Under Type-II Censoring Scheme,” Computational Statistics and Data Analysis, Vol. 48, No. 3, 2005, pp. 509-523. doi:10.1016/j.csda.2004.02.009

[18] L. F. Zhang, M. Xie and L. C. Tang, “A Study of Two Estimation Approaches for Parameters of Weibull Distribution Based on WPP,” Reliability Engineering and System Safety, Vol. 92, No. 3, 2007, pp. 360-368. doi: 10.1016/j.ress.2006.04.008

[19] Z. F. Jaheen, “Bayesian Prediction Under a Mixture of Two-Component Gompertz Lifetime Model,” Test, Vol. 12, No. 1, 2003, pp. 413-426. doi:10.1007/BF02595722

[20] M. Zhou, “Asymptotic Normality of the Synthetic Data Regression Estimation for Censored Survival Data,” Annals of Statistics, Vol. 20, No. 2, 1992, pp. 1002-1021. doi:10.1214/aos/1176348667

[1] H. Robbins, “Some Thoughts on Empirical Bayes Estimation,” Annals of Statistics, Vol. 11, No. 3, 1983, pp. 713-723. doi:10.1214/aos/1176346239

[2] S. Zacks, “Introduction to Reliability Analysis,” Springer, New York, 1991.

[3] J. Aitchison and I. R. Dunsmore, “Statistical Prediction Analysis,” Cambridge University Press, Cambridge, 1975. doi:10.1017/CBO9780511569647

[4] S. Geisser, “Predictive Inference: An Introduction,” Chapman and Hall, London, 1993.

[5] L. J. Bain and M. Engelhardt, “Interval Estimation for the Two-Parameter Double Exponential Distribution,” Technometrics, Vol. 15, No. 4, 1973, pp. 875-887. doi: 10.2307/1267397

[6] R. G. Easterling, “Exponential Responses with Double Exponetial Distribution Measurement Error-A Model for Steam Generator Inspection,” Proceedings of the DOE Statistical Symposium, U.S. Department of Energy, 1978, pp. 90-110.

[7] M. T. Madi and T. Leonard, “Bayesian Estimation for Shifted Exponential Distribution,” Journal of Statistical Planning and Inference, Vol. 55, No. 6, 1996, pp. 345-351. doi:10.1016/S0378-3758(95)00199-9

[8] W. T. Huang and H. H. Huang, “Empirical Bayes Estimation of the Guarantee Lifetime in a Two-Parameter Exponential Distribution,” Statistics and Probability Letters, Vol. 76, No. 16, 2006, pp. 1821-1829. doi:10.1016/ j.spl.2006.04.034

[9] J. W. Wu, H. M. Lee and C. L. Lei, “Computational Testing Algorithmic Procedure of Assessment for Lifetime Performance Index of Products with Two-Parameter Exponential Distribution,” Applied Mathematics and Computation, Vol. 190, No. 5, 2007, pp. 116-125. doi: 10.1016/j.amc.2007.01.010

[10] J. O. Berger, “Statistical Decision Theory and Bayesian Analysis,” Springer, New York, 1985.

[11] R. S. Singh, “Bayes and Empirical Bayes Procedures for Selecting Good Populations from a Translated Exponential Family,” Empirical Bayes and Likelihood Inference, Springer, New York, 2001.

[12] H. Lian, “Consistency of Bayesian Estimation of a Step Function,” Statistics and Probability Letters, Vol. 77, No. 1, 2007, pp. 19-24. doi:10.1016/j.spl.2006.05.007

[13] Q. H. Wang, “Empirical Likelihood for a Class of Functions of Survival Distribution with Censored Data,” Annals of the Institute of Statistical Mathematics, Vol. 53, No. 3, 2001, pp. 517-527. doi:10.1023/A:1014617112870

[14] E. T. Kaplan and P. Meier, “Nonparametric Estimation from Incomplete Observations,” Journal of the American Statistical Association, Vol. 53, No. 282, 1958, pp. 457-481. doi:10.2307/2281868

[15] R. Y. Zheng, “Improvement on Grey Modelling Approach and Its Applications to Fatigue Reliability Research,” Master Dissertation, National University of Defense Technology, Chang Sha, 1989.

[16] K. O. Bowman and L. R. Shenton, “Weibull Distributions When the Shape Parameter is Defined,” Computational Statistics and Data Analysis, Vol. 36, No. 4, 2001, pp. 299-310. doi:10.1016/S0167-9473(00)00048-7

[17] U. Singh, P. K. Gupta and S. K. Upadhyay, “Estimation of Parameters for Exponentiated-Weibull Family Under Type-II Censoring Scheme,” Computational Statistics and Data Analysis, Vol. 48, No. 3, 2005, pp. 509-523. doi:10.1016/j.csda.2004.02.009

[18] L. F. Zhang, M. Xie and L. C. Tang, “A Study of Two Estimation Approaches for Parameters of Weibull Distribution Based on WPP,” Reliability Engineering and System Safety, Vol. 92, No. 3, 2007, pp. 360-368. doi: 10.1016/j.ress.2006.04.008

[19] Z. F. Jaheen, “Bayesian Prediction Under a Mixture of Two-Component Gompertz Lifetime Model,” Test, Vol. 12, No. 1, 2003, pp. 413-426. doi:10.1007/BF02595722

[20] M. Zhou, “Asymptotic Normality of the Synthetic Data Regression Estimation for Censored Survival Data,” Annals of Statistics, Vol. 20, No. 2, 1992, pp. 1002-1021. doi:10.1214/aos/1176348667