Approximate Bayesian Inferences for the Reliability in the Weibull Case: Some Aspects of Reparametrization

Abstract

Usually, inferences on the reliability function R(t ₀) at time t ₀ assuming different parametrical models and censored lifetime observations are obtained by using asymptotical methods. One of these asymptotical results is given by the asymptotical normality of the maximum likelihood estimators. Under the Bayesian approach, we get marginal posterior densities or posterior moments for R(t ₀) based on numerical or approximation methods. These results, usually depend on an appropriate transformation of R(t ₀ ), to get accurate results. One way to find an appropriate reparametrization, is to search for a one-to-one transformation of R(t ₀) that gives close “normality” for the likelihood function (see for example, Anscombe, 1964; Sprott, 1973, 1980; Kass and Slate, 1992; or Hills and Smith, 1993). Assuming a Weibull distribution for the lifetimes in a reliability experiment, we explore the use of some popular transformations for proportions (see for example, Guerrero and Johnson, 1982; or Aranda-Ordaz, 1981) and a measure to nonnormality of likelihood functions or posterior densities given by the standardized form of the third derivative of the logarithm of the likelihood or posterior density (see Sprott, 1973; or Kass and Slate, 1992). We also check the adequability of the proposed reparametrization, by using the t-plot proposed by Hills and Smith (1993).