A simple procedure for Bayesian estimation of the Weibull distribution

Practical use of Bayesian estimation procedures is often associated with difficulties related to elicitation of prior information, and its formalization into the respective prior distribution. The two-parameter Weibull distribution is a particularly difficult case, because it requires a two-dimensional joint prior distribution of the Weibull parameters. The novelty of the procedure suggested here is that the prior information can be presented in the form of the interval assessment of the reliability function (as opposed to that on the Weibull parameters), which is generally easier to obtain. Based on this prior information, the procedure allows constructing the continuous joint prior distribution of Weibull parameters as well as the posterior estimates of the mean and standard deviation of the estimated reliability function (or the CDF) at any given value of the exposure variable. A numeric example is discussed as an illustration. We additionally elaborate on a new parametric form of the prior distribution for the scale parameter of the exponential distribution. This distribution is not a Gamma (as might intuitively be expected); its mode is available in a closed form, and the mean is obtainable through a series approximation.