Bayesian Lower Bounds on Reliability for the Lognormal Model

Bayesian lower bounds for the reliability function are obtained for the lognormal failure model with respect to the s-normal-gamma (conjugate) prior distribution and a vague prior distribution of Jeffreys. The Bayesian lower bound with respect to the vague prior is the same as the uniformly most accurate (UMA) lower s-confidence bound for reliability. All lower bounds are given in terms of the noncentrality parameter of a generalized noncentral t-distribution. A simple approximation for the noncentrality parameter is discussed. Computer simulation results indicate how well the approximation performs and provide a performance comparison between the Bayes lower bounds with respect to the (proper) s-normal-gamma prior and the UMA lower s-confidence bound. The two measures used in the simulations to evaluate performance of the lower bounds are 1) the average difference between the computed lower bound and the true reliability and 2) the fraction of computed lower bounds which are actually less than the true reliability. This Bayes procedure performs very well even though the assumed prior information is not exactly correct; and the approximation is used to obtain the lower bounds.