Bayesian Copula Models Based On MCMC And Application Research

Dependent structure analysis is an important topic in the field of reliability engineering, survival analysis and financial research, and so on, which has attracted considerable interest and is more valuable, product quality monitoring, life characteristic analysis, financial market investment portfolio, risk aversion and asset management all need to consider the dependence structure in different variables. Copulas functions are the statistical tools to characterize non-normal, asymmetric, nonlinear and dynamic dependence structures of variables. Under the condition of continuous, discrete and censored variable, the paper combines the Bayesian inference theory and the Copula functions theory, to explore the dependence structure in the areas of reliability, survival analysis and finance; and to estimate the parameters of marginal and joint distribution based on MCMC sampling algorithms; to compare the advantages and disadvantages of different parameter estimation methods; to research the application of the related model based on simulation and empirical analysis.First, Frank copula reliability model are considered combined copula function and exponential and Pareto distribution, including joint distribution and probability density function is derived, the sampling algorithm for the marginal distribution parameter. The sampling parameter estimation procedure is constructed based on MCMC sampling algorithms, including the determination of super parameters and covariance matrix, the design Metropolis-Hastings sampling algorithm for two types of Frank copula models is given. The Bayesian estimation results of parameters in exponential Frank copula model is presented based on simulation, at the same time, the effectiveness and robustness of the results are tested using the Bayesian p statistics, and experimental results show the accurate of the algorithm proposed.Secondly, the Bayesian inference theory for copula survival model based on censored variable is explored, including frailty、positive stable and cured survival model. The conditional posterior distribution of frailty survival model parameters are presented; and the estimated results for marginal parameters are discussed using Gibbs sampling, adaptative and M-H sampling algorithm respectively; the conditional posterior of dependence parameters are discussed based on two-step and one-step Bayesian methods, and the completely conditional posterior distributions for cured copula survival based on censored variable are also presented using Gibbs sampling. Using the real censored data, the statistics for posterior dependence parameters of two-step and one-step Bayesian estimations are discussed under positive stable、Frank and Clayton copula censored survival model respectively, and to compare the copula models using popular Bayesian selection criteria, such as the DIC、EAIC、EBIC and CPO statistic.Furthermore, the Bayesian inference and estimate for multivariate copula model are discussed when the marginal distribution variables are continuous, discrete and mixed case respectively, the correlation matrix is parameterize using binary indicator variables, thus, the estimations for the latent variable and parameterized matrix elements are discussed based on M-H sampling algorithm, at the same time, the Bayesian inference for multivariate copula models are explored for discrete and mixed variables, and conditional posterior distribution of parameters are presented for marginal distribution, latent variables and dependence using MCMC sampling respectively. The multivariate copula regression model and the prior choice for covariance matrix are discussed, and the MCMC sampling algorithms for marginal distribution parameters and the element for correlation matrix are given. The MCMC sampling process of normal copula for mixed variable are achieved based on simulation, and discuss the estimation and test for the related parameters of model.Finally, the Bayesian inference for time-varying t-copula model is explored based on time series. The dependence structure between crude oil prices and stock markets in the Asia-Pacific region are discussed before and after of the global financial crisis based on the constant, time-varying and Bayesian time-varying copula model respectively. The results show that the dependence increased significantly in the aftermath of the crisis, and time-varying copulas models capture the dependence structure better than constant models, and the VaR estimate of portfolio for the crude oil and the Asia-Pacific stock are proposed using three kinds of copula models, the time-varying Bayesian copulas provide better VaR estimates。...