Bayesian Estimation of Life Parameters in the Weibull Distribution

This paper develops a Bayesian analysis of the scale and shape parameters in the Weibull distribution and the corresponding reliability function with respect to the usual life-testing procedures. For the scale parameter θ, Bayesian estimates of θ and reliability are obtained for the uniform, exponential, and inverted gamma prior probability densities. Bhattacharya's results [J. Am. Stat. Assn. 62, 48–62 (1967)] for the one-parameter exponential life-testing distribution are reduced to a special case of these results. The paper develops a fully Bayesian analysis of both the scale and shape parameters θ and ξ by assuming independent prior distributions; since in the latter case, analytical tractability is not possible, Bayesian estimates are obtained through a conjunction of a Monte Carlo simulation and numerical-integration techniques. In both cases, the paper carries out a computer simulation and makes a comparison between the Bayesian and the corresponding minimum-variance unbiased, or maximum likelihood, estimates. As expected, the Bayesian estimates are superior.