BAYESIAN SEQUENTIAL RELIABILITY FOR WEIBULL AND RELATED DISTRIBUTIONS

DONGCHU SUN^{1} AND JAMES O. BERGER^{2}

1Department of Statistics, University of Missouri-Columbia, Columbia, MO 65211, U.S.A. 2Department of Statistics, Purdue University, West Lafayette, IN 47907, U.S.A.

(Received July 10, 1992; revised June 1, 1993)

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 Exponential E(θ), Rayleigh, Weibull W(θ, β) and Pareto pdf’s are special cases. Q(θ) will be assumed to have an inverse Gamma prior. Assume that m 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.

Key words and phrases: Bayesian sequential test, reliability demonstration test, exponential distribution, Weibull distribution, expected stopping time, producer’s risk, consumer’s risk.