Bayesian Estimation of a Weibull distribution in a highly censored and small sample setting

We propose and investigate through Monte Carlo simulations two methods for Bayesian inference for the shape and the scale parameters of a Weibull distribution in a small and highly censored sample setting. The rst method, WLB-SIR, is the Sampling Importance Resampling-adjusted Weighted Likelihood Bootstrap of Newton and Raftery. The second one is a new Bayesian Restoration Maximization (BRM) new algorithm working along the same line but replacing the bootstrap weighting step by a stochastic simulation step of the censored failure times. The advantage of the BRM method is that it takes account of the prior distribution in its rst step. As a consequence, the Sampling Importance Resampling-adjusted version of BRM, BRM-SIR, is less fragile than WLB-SIR as it appears from our numerical experiments for Weibull parameters estimation. This article also includes a exible procedure to transform prior knowledge into prior distributions on the Weibull parameters. Estimation bay esienne des param etres d'une loi de Weibull pour de petits echantillons tr es censur es. R esum e : Nous proposons deux m ethodes d'inf erence bay esienne pour estimer le param etre de forme et d' echelle d'une loi de Weibull pour de petits echan-tillons tr es censur es. Nous analysons leurs performances par des simulations de Monte Carlo. Ces deux m ethodes sont le Weighted Likelihood Bootstrap (WLB) de Newton et Raftery et une nouvelle m ethode Bayesian Restoration Maximization (BRM) qui travaille dans le m^ eme esprit, mais remplace les ti-rages de poids a l'aide du bootstrap par la simulation des donn ees censur ees. L'avantage de BRM sur WLB est qu'il tient compte dans sa phase initiale de la loi a priori des param etres. Ainsi, il fournit des r esultats plus ables et plus stables comme le montrent nos exp erimentations num eriques. Cet article propose de plus une proc edure souple pour traduire les simples connaissances a priori en distributions a priori.