Robust Bayesian Inference In Multivariate Heteroscedastic Replicated Measurement Error Models

Due to the inaccuracy of the measuring instrument, the instability of the mea-surement environment, the measurement error of the measurement personnel and other factors, the measurement value of the interest variable is often inconsistent with the real one. There exists measurement error in the scientific experiment, the production or the statistical investigations, so the measurement error model analysis and data pro-cessing are of great significance. At the same time, the heteroscedasticity phenomenon,which means the variance of the measurement error is not static but changing with the characteristics of the sample individuals, can not be ignored in statistical inference,otherwise it will seriously affect the accuracy of statistical results. Based on the multi-variate measurement error model, this paper discusses the statistical inference problem of the multivariate heteroscedastic measurement error model for replicated data with and without the equation error, and gives the parameter estimation method under the Bayesian framework. Through statistical simulation and instance analysis for global root decomposition data and Egyptian ceramic data, verifying the effectiveness of the model and the efficiency of the proposed method.The main work of the master's thesis is as follows:Chapter 1 introduces the research background and research status of the mea-surement error model, the basic concepts and properties of the scale mixtures of nor-mal distribution, the basic principle of the Bayesian inference and the model selection method used in this paper.In chapter 2, we study the parameter estimation of the multivariate heteroscedastic measurement error model for replicated data with equation error under heavy-tailed distributions. Firstly, the structure of the model is proposed. Secondly, the MCMC algorithm is used to estimate the parameters of the model. Thirdly, two numerical simulations show that heavy-tailed distribution models are more robust than the normal distribution in the case of distribution misclassification and data anomalies. Finally,the analysis of global root decomposition data verifies the validity of Bayesian inference under this model.In chapter 3, based on the theory of the previous chapter, we study the parameter estimation of the multivariate heteroscedastic measurement error model for replicated data with no equation error under heavy-tailed distributions. Firstly, the structure of the model is proposed. Secondly, the detailed MCMC algorithm of parameter estima-tion is given. Finally, through the case analysis of Egyptian ceramic data, the model with the best fit is selected.In chapter 4, we study the sensitivity analysis and influence analysis of the param-eter estimation of the multivariate heteroscedastic measurement error model for repli-cated data with equation error under heavy-tailed distributions in Bayesian framework.Through the numerical simulation and the instance analysis, we show the importance of the selection of prior distributions and the hyper parameter. We also show the influ-ence of the outliers on the parameter estimation, and once again prove the robustness of the heavy-tailed distribution model facing the data with outliers.Chapter 5 summarizes the series of models and methods discussed in this paper,and puts forward the direction that can be further studied.