基于MCMC的贝叶斯Copula模型构建及应用研究

发布时间：2018-04-01 21:44

本文选题：贝叶斯推断　切入点：Copula函数　出处：《湖南大学》2014年博士论文

【摘要】：相依结构分析是可靠性工程、生存分析和金融等领域的重要研究问题,产品的质量监控、寿命特征分析、金融市场投资组合、风险回避和资产管理等都需要考虑变量间相依结构。Copula函数是刻画变量非正态、非对称、非线性和动态等相关关系的统计工具。本文利用贝叶斯推断理论结合Copula函数方法,探讨变量类型分别为连续、离散、混合和删失等情形下,研究可靠性、生存分析和金融等领域描述变量相依结构的Copula贝叶斯建模理论,设计边际和相依结构参数的MCMC抽样算法,比较不同参数估计方法的优劣,仿真和实证研究所构建模型在可靠性、生存分析和金融等领域的应用。首先,利用Copula函数理论结合指数和Pareto分布构建了Frank Copula可靠性模型,包括联合分布函数、概率密度函数的推导和边际分布参数的抽样算法；利用MCMC抽样理论构造了参数的估计过程,包括超参数的确定、参数协方差矩阵的设定和两类Frank Copula模型参数的M-H抽样算法；通过仿真分析给出指数Frank Copula模型参数的贝叶斯估计结果,利用贝叶斯p统计量检验估计的有效性和稳健性,结果表明贝叶斯估计能准确估计参数。然后,研究了基于删失数据的Copula生存模型的贝叶斯推断理论。包括异质、正稳态和治愈率删失Copula生存模型构建；推导异质删失Copula生存模型参数的条件后验分布；设计Gibbs抽样、自适宜和M-H抽样算法对正稳态删失Copula生存模型边际参数的估计；利用一步和两阶段贝叶斯估计分别推导相依参数的条件后验分布；设计Gibbs抽样推导治愈率删失Copula生存模型参数的完全条件后验分布。利用删失生存的实际数据,分别用删失正稳态、Frank和Clayton Copula生存模型估计变量间相依结构,给出两阶段与一步贝叶斯估计的参数后验统计量,然后利用DIC、EAIC、EBIC和CPO等统计量对所用模型进行比较选择分析。其次,研究了贝叶斯方法对边际分布为连续、离散和混合变量的多元Copula模型参数估计和统计推断理论。引入二元指示变量对相关矩阵参数化,设计M-H抽样算法完成连续多元Copula模型的潜变量和参数化矩阵元素的估计。讨论离散和混合变量的多元Copula模型构建,利用MCMC抽样得到边际分布、潜变量和相依参数的条件后验分布。构建多元Copula回归模型,讨论协方差矩阵的先验选择,研究离散和混合变量情形下边际分布参数和相关矩阵元素的MCMC抽样过程。同时结合Monte Carlo仿真对混合变量的正态Copula模型的贝叶斯抽样过程进行实现,给出相关参数的后验估计和检验。最后,研究了基于时间序列的时变t-Copula模型的贝叶斯推断理论。利用静态Copula、时变Copula和时变Copula贝叶斯模型分别描述金融危机前后国际原油价格与亚太股票市场的相依结构。研究结果表明,金融危机后相依结构比危机前明显增强,时变Copula模型更加适合刻画变量间的相依结构,同时利用静态Copula、时变Copula和时变Copula贝叶斯模型估计原油与亚太股票市场投资组合的VaR,发现时变t-Copula贝叶斯模型可以更好地估计投资组合的VaR。
[Abstract]:According to the theory of Bayesian inference and Copula ' s function method , we study the research reliability , the survival analysis and the financial and so on . The Copula function is the continuous , discrete , mixed and censored data statistical tool .

Firstly , the Frank Copula reliability model is constructed by using the combination index and Pareto distribution of the Copula function , including the joint distribution function , the derivation of the probability density function and the sampling algorithm of the marginal distribution parameter ;
The estimation process of parameters is constructed by MCMC sampling theory , including the determination of superparameter , the setting of parameter covariance matrix and the M - H sampling algorithm of two kinds of Frank Copula model parameters .
The Bayesian estimation results of the index Frank Copula model parameters are given by means of simulation analysis , and the validity and robustness of the estimation are verified by using the Bayesian p statistics , and the results show that the Bayesian estimation can accurately estimate the parameters .

Then , the Bayesian inference theory of Copula survival model based on censored data is studied , including heterogeneous , positive steady state and cure rate censored Copula survival model .
deriving the conditional posterior distribution of the parameters of the heterogeneous censored Copula survival model ;
Design Gibbs Sampling , Adaptive and M - H Sampling Algorithm for Estimation of Marginal Parameters of Positive Steady State censored Copula Survival Model ;
using one - step and two - stage Bayesian estimation to derive the conditional posterior distribution of the dependent parameters respectively ;
Gibbs sampling was used to derive the complete conditional posterior distribution of the parameters of the Copula survival model .

Using censored real data , we estimate the dependent structure of variables by censored n - steady state , Frank and the Copula survival model , and then give a statistical measure of the parameters after two - stage and one - step Bayesian estimation . Then , the model is selected and analyzed by using statistics such as DIC , EAIC , EBIC and CPO .

Secondly , the multivariate Copula model parameter estimation and statistical inference theory for the continuous , discrete and mixed variables of the Bayesian method are studied . The multivariate Copula model of discrete and mixed variables is introduced . The multivariate Copula model of discrete and mixed variables is designed , the marginal distribution parameters and the MCMC sampling process of the correlation matrix elements are obtained by using MCMC sampling . The Bayesian sampling process of the positive Copula model of the mixed variable is studied by Monte Carlo simulation . The posterior estimation and the inspection of the parameters are given .

Finally , we study the Bayesian inference theory of time - varying t - Copula model based on time series . By using the static Copula , the time - varying Copula and the time - varying Copula Bayes model , we describe the dependence structure of the international crude oil price before and after the financial crisis and the Asia - Pacific stock market respectively .

【学位授予单位】：湖南大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：F830.91;F224

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