Objective Bayesian Analysis For Accelerated Degradation Models

In the theory of reliability,the phenomenon of loss the specified function of a product is often referred to the failure.However,with the rapid progress and development of science and technology,the reliability of the product has been greatly improved.Due to many high reliability products often do not fail in a few years or even decades,it brings great difficulties and challenges for us to collect failure data of products.However,most products have one or more quality characteristics,which will degrade as time goes on,and eventually lead to the failure of the product.Therefore,we can predict the reliability of the product by studying the degradation data of these quality characteristics.However,due to the high reliability of the product,it is sometimes difficult to collect suffient degradation data under normal experimental conditions.As we know,the degradation of a product is often influenced by many environmental factors or experimental conditions,such as temperature,voltage,humidity,pressure and so on.These factors are usually called acceleration variables or stress levels.Therefore,an alternative effective method is to collect degradation data of the product at the high stress levels,and then evaluate the reliability of the product.On the other hand,in recent years,the objective Bayesian method has attracted more and more attention due to its many advantages.This paper aim to study the accelerated degradation data from the perspective of objective Bayesian statistics.In this paper,we first introduce the objective Bayesian theory,and focus on two kinds of noninformative prior,which are the Jeffreys prior and the reference prior.Then,we consider the accelerated degradation model based on the Wiener process with measurement errors and the accelerated degradation model based on the inverse Gaussian process,respectively.For these two models,we use the objective Bayesian method to derive the Jeffreys prior and reference priors under different group orderings.At the same time,we theoretically prove the propriety of the posterior distribution under each prior.The simulation results show that the objective Bayesian method is better than the maximum likelihood estimation and the Bootstrap method in the light of the mean square error and the frequentist coverage probability.Furthermore,we apply the proposed method to real data sets and assess the reliability of the product.At the end of the paper,we briefly summarize our research work and point out some problems that can be further studied.