Abstract
Assume that the probability density function for the lifetime of a newly designed product has the form: [H′(t)/Q(θ)] exp{−H(t)/Q(θ)}. The Exponentialε(θ), Rayleigh, WeibullW(θ, β) and Pareto pdf's are special cases.Q(θ) will be assumed to have an inverse Gamma prior. Assume thatm independent products are to be tested with replacement. A Bayesian Sequential Reliability Demonstration Testing plan is used to eigher accept the product and start formal production, or reject the product for reengineering. The test criterion is the intersection of two goals, a minimal goal to begin production and a mature product goal. The exact values of various risks and the distribution of total number of failures are evaluated. Based on a result about a Poisson process, the expected stopping time for the exponential failure time is also found. Included in these risks and expected stopping times are frequentist versions, thereof, so that the results also provide frequentist answers for a class of interesting stopping rules.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balaban, H. S. (1975). Reliability demonstration: purposes, practics, and value,Proceedings 1975 Annual Reliability and Maintainability Symposium, 246–248, The Institute of Electrical and Electronics Engineers, California.
Barlow, R. E. and Campo, R. (1975). Total time on test processes and application to failure data analysis,Reliability and Fault Tree Analysis (eds. R. E. Barlow, J. Fussell and N. D. Singpurwalla), 451–481, SIAM, Philadelphia.
Barnett, V. D. (1972). A Bayes sequential life test,Technometrics,14, 453–467.
Billingsley, P. (1986).Probability and Measure, 2nd ed., Wiley, New York.
Bivens, G., Born, F., Caroli, J. and Hyle, R. (1987). Reliability demonstration technique for fault tolerant systems,Proceedings 1987 Annual Reliability and Maintainability Symposium, 316–320, The Institute of Electrical and Electronics Engineers, California.
Bremaud, P. (1981).Point Processes and Queues, Martingale Dynamics, Springer, New York.
Chandra, M. and Singpurwalla, N. D. (1981). Relationships between some notions which are common to reliability theory and economics,Math. Oper. Res.,6, 113–121.
Easterling, R. G. (1970). On the use of prior distribution in acceptance sampling,Annals of Reliability and Maintainability,9, 31–35.
Eastering, R. G. (1975). Risk quantification,Proceedings 1975 Annual Reliability and Maintainability Symposium, 249–250, The Institute of Electrical and Electronics Engineers, California.
Epstein, B. and Sobel, M. (1953). Life testing,J. Amer. Statist. Assoc.,48, 486–502.
Epstein, B. and Sobel, M. (1955). Sequential life tests in the exponential case,Ann. Math. Statist.,26, 82–93.
Goel, A. L. (1975). Panel discussion on some recent developments in reliability demonstration, introductory remarks,Proceedings 1975 Annual Reliability and Maintainability Symposium, 224–245, The Institute of Electrical and Electronics Engineers, California.
Goel, A. L. and Coppola, A. (1979). Design of reliability acceptance sampling plans based upon prior distribution,Proceedings 1979 Annual Reliability and Maintainability Symposium, 34–38, The Institute of Electrical and Electronics Engineers, California.
Goel, A. L. and Joglekar, A. M. (1976). Reliability acceptance sampling plans based upon prior distribution, Tech. Reports, 76-1 to 76-5, Dept. of Industrial Engineering and Operations Research, Syracuse University, Syracuse, New York.
Harris, C. M. and Singpurwalla, N. D. (1968). Life distributions denied from stochastic hazard functions,IEEE Transitions on Reliability,R-17, 70–79.
Harris, C. M. and Singpurwalla, N. D. (1969). On estimation in Weibull distribution with random scale parameters,Naval Res. Logist. Quart.,16, 405–410.
Jewel, W. S. (1977). Bayesian life testing using the totalQ on test,The Theory and Application of Reliability, with Emphasis on Bayesian and Nonparametric Method (eds. C. P. Tsokos and I. N. Shimi), 49–66, Academic Press, New York.
Klefsjö, B. (1991). TTT-plotting—a tool for both theoretical and practical problems,J. Statist. Plann. Inference,29, 99–110.
Lindley, D. V. and Singpurwalla, N. D. (1991a). On the amount of evidence needed to reach agreement between adversaries,J. Amer. Statist. Assoc.,86, 933–937.
Lindley, D. V. and Singpurwalla, N. D. (1991b).Adversarial life testing (submitted).
MacFarland, W. J. (1971). Sequential analysis and Bayes demonstration,Proceedings 1971 Annual Reliability and Maintainability Symposium, 24–38, The Institute of Electrical and Electronics Engineers, California.
Mann, N. R., Schafer, R. E. and Singpurwalla, N. D. (1974).Methods for Statistical Analysis of Reliability and Life Data, Wiley, New York.
Martz, H. F. and Waller, R. A. (1979). A Bayesian zero-failure (BAZE) reliability demonstration testing procedure,Journal of Quality Technology,11, 128–138.
Martz, H. F. and Waller, R. A. (1982).Bayesian Reliability Analysis, Wiley, New York.
Montagne, E. R. and Singpurwalla, N. D. (1985). Robustness of sequential exponential life-testing procedures,J. Amer. Statist. Assoc.,80, 715–719.
Ray, S. M. (1965). Bounds on the minimum sample size of a Bayes sequential procedure,Ann. Math. Statist.,39, 859–878.
Schafer, R. E. (1969). Bayesian reliability demonstration, phase I—data for the a priori distribution, RADC-TR-69-389, Rome Air Development Center, Rome, New York.
Schafer, R. E. (1975). Some approaches to Bayesian reliability demonstration,Proceedings 1975 Annual Reliability and Maintainability Symposium, 253–254, The Institute of Electrical and Electronics Engineers, California.
Schafer, R. E. and Sheffield, T. S. (1971). Bayesian reliability demonstration, phase II—development of a prior distribution, RADC-TR-71-139, Rome Air Development Center, Rome, New York.
Schafer, R. E. and Singpurwalla, N. D. (1970). A sequential Bayes procedure for reliability demonstration,Naval Res. Logist. Quart.,17, 55–67.
Schemee, J. (1975). Application of the sequentialt-test to maintainability demonstration,Proceedings 1975 Annual Reliability and Maintainability Symposium, 239–243, The Institute of Electrical and Electronics Engineers, California.
Schick, G. J. and Drnas, T. M. (1972). Bayesian reliability demonstration,AIIE Transactions,4, 92–102.
Soland, R. M. (1968). Bayesian analysis of the Weibull process with unknown scale parameter and its application to acceptance sampling,IEEE Transactions on Reliability,R-17, 84–90.
Soland, R. M. (1969). Bayesian analysis of the Weibull process with unknown scale parameter and shape parameters,IEEE Transactions on Reliability,R-18, 181–184.
Author information
Authors and Affiliations
Additional information
This research was supported by NSF grants DMS-8703620 and DMS-8923071, and forms part of the Ph.D. Thesis of the first author, the development of which was supported in part by a David Ross grant at Purdue University. The authors thank the editors and a referee for insightful comments and suggestions.
About this article
Cite this article
Sun, D., Berger, J.O. Bayesian sequential reliability for Weibull and related distributions. Ann Inst Stat Math 46, 221–249 (1994). https://doi.org/10.1007/BF01720582
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01720582