## AbstractThe main issues raised by the estimation of lifetime parametric models used in industrial modelling of durability are censoring and REX sample size. Many studies of durability must take into account homogeneous, small-sized, censored failure times which have to be integrated into Bayesian procedures with informative prior parameter. This way of dealing with statistical inference has been especially chosen by EDF for predicting failures on nuclear material. Firstly experts have to be questioned about the durability of a material with precise understandable questions. According to the choice of the considered model, prior point estimations and confidence intervals about parameters must be given directly or indirectly by experts. Secondly efficient modelling has to be chosen for informative prior distributions. Once, it must produce posterior distributions easily estimated by classical methods (conjugate distributions if possible, MCMC algorithms as Gibbs sampling, or importance sampling). But computation complexity is often a limiting factor of Bayesian inference. The impact of prior choices on posterior results must be simple to derive. Then, hyperparameters of these prior distributions must be evaluated linking the intrinsic properties of the prior densities (mean, mode, variance, etc.) with expert information on parameters. The example of the Weibull distribution will be thoroughfully treated. Denoting W(η,β) the Weibull distribution with scale parameter β and shape parameter η (homogeneous to the lifetime), the parameters of this standard failure time model can be easily linked to expert decision. Thus, some precise questions on aging behaviour and failures times will be proposed, which were assessed by EDF experts, in order to obtain enough information to create informative prior distribution on η and a uniform prior distribution on β. Choosing Generalized Inverse Gamma distributions GIG(a,b,β) as prior distribution on η will help to work easier with a reparametrization of the Weibull distribution, denoted W(μ,β) with μ=η-β. This modelling is firstly well adapted to sampling methods. Indeed, knowing any prior estimation of β, μ will follow a simple Gamma G(a,b) prior distribution. Thus the posterior distribution of μ will be a Gamma distributions G(a+n,b+t) where t depends of the real data (of size n) and posterior estimations of β can be computed with sampling methods. Thus prior choices can be compared with standard statistics. Secondly this prior modelling facilitate expert opinion combining by successive Bayesian inferences with fictitious data sets associated to each expert opinion. Hyperparameters are computed to find the best fit between empirical prior distributions and knowledge experts. Hyperparameters (a, b) are selected my minimizing criteria measuring the adequacy between expert knowledge and simulated prior distributions. Finally a numerical example with real lifetime data from EDF and two different experts from EDF and an external analyst will be presented.
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