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BAYESIAN SEQUENTIAL RELIABILITY FOR WEIBULL

AND RELATED DISTRIBUTIONS

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

^{1} *Department of Statistics, University of Missouri-Columbia,
Columbia, MO 65211, U.S.A.*

^{2} *Department 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*(*theta*)]exp{-*H*(*t*)/*Q*(*theta*)}. The Exponential \cal
E(*theta*), Rayleigh, Weibull \cal W(*theta*, *beta*) and Pareto
pdf's are special cases. *Q*(*theta*) 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.

**Source**
( TeX ,
DVI ,
PS )