Bayesian sequential reliability for Weibull and related distributions

Assume that the probability density function for the lifetime of a newly designed product has the form: $\lbrack H\prime(t)/Q(\theta)\rbrack $exp$\{-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 either 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. Also, a discrete BSRDT plan is discussed. The exact forms of the various risks the distributions of the number of failures and sample size are evaluated. If only one unit at a time tested, the expected testing time is given in closed form. Some asymptotic properties of the discrete BSRDT plan are also discussed. Under suitable conditions, the discrete BSRDT plan is asymptotically equivalent to the Bayes sequential test. A stepwise testing model generalizes the testing model. A predictive BSRDT plan is proposed, which is based on the predictive distribution of a future observation. Various interesting quantities mentioned above can be evaluated as well. Bayesian analysis for a Poly-Weibull p.d.f. under some proper and noninformative priors are discussed. A recursive formula to compute the relative posterior p.d.f. will be developed. This formula generalizes a result by Berger(1990), and reduces the total number of computations for Poly-Webull distributions from $m\sp{n}$ to ${m + n\choose m}.$