Bayesian Statistical Diagnostics In Linear Models And Degradation Models

In this paper,we investigate some issues in linear models and degradation models based on Bayesian paradigm.Specifically,Bayesian estimation is studied in multivariate linear model and the theoretical conditions required for proper posterior are obtained under improper hierarchical priors.In addition,we propose an objective Bayesian method to analyze the degradation model based on Wiener process.In the first chapter,we introduce some basic concepts in Bayesian statistics,and discuss the calculation methods of Bayesian statistical diagnostics and its application in reliability analysis.In chapter two,we investigate the estimation problem of the parameters in the multivariate linear model while the design matrix is rank-deficient.The joint prior of regression coefficient matrix and covariance matrix is assumed to be normal-inverse Wishart distribution.The Bayesian estimation of estimable function of regression coefficient matrix and covariance matrix are then derived.Furthermore,the superiority of these Bayesian estimators is studied.In the third chapter,the theoretical conditions for which improper hierarchical priors can yield proper posteriors in the multivariate linear model are derived.The importance of the theoretical results are then illustrated by simulation studies and real data analysis.In chapter four,we propose an objective Bayesian method to study the degra-dation model with respect to a Wiener process.The Jeffreys prior and reference prior of the parameters are derived,and the propriety of the posteriors under these priors is validated.Two sampling algorithms are introduced to compute the pos-teriors.A simulation study is conducted to assess the performance of the objective Bayesian procedure.Besides,the approach is applied to analyze a degradation data.Finally,we briefly summarize our research work and point out some problems which deserve further study.