本文選題：貝葉斯推斷　切入點：Copula函數(shù)　出處：《湖南大學(xué)》2014年博士論文

【摘要】：相依結(jié)構(gòu)分析是可靠性工程、生存分析和金融等領(lǐng)域的重要研究問題,產(chǎn)品的質(zhì)量監(jiān)控、壽命特征分析、金融市場投資組合、風(fēng)險回避和資產(chǎn)管理等都需要考慮變量間相依結(jié)構(gòu)。Copula函數(shù)是刻畫變量非正態(tài)、非對稱、非線性和動態(tài)等相關(guān)關(guān)系的統(tǒng)計工具。本文利用貝葉斯推斷理論結(jié)合Copula函數(shù)方法,探討變量類型分別為連續(xù)、離散、混合和刪失等情形下,研究可靠性、生存分析和金融等領(lǐng)域描述變量相依結(jié)構(gòu)的Copula貝葉斯建模理論,設(shè)計邊際和相依結(jié)構(gòu)參數(shù)的MCMC抽樣算法,比較不同參數(shù)估計方法的優(yōu)劣,仿真和實證研究所構(gòu)建模型在可靠性、生存分析和金融等領(lǐng)域的應(yīng)用。 首先,利用Copula函數(shù)理論結(jié)合指數(shù)和Pareto分布構(gòu)建了Frank Copula可靠性模型,包括聯(lián)合分布函數(shù)、概率密度函數(shù)的推導(dǎo)和邊際分布參數(shù)的抽樣算法；利用MCMC抽樣理論構(gòu)造了參數(shù)的估計過程,包括超參數(shù)的確定、參數(shù)協(xié)方差矩陣的設(shè)定和兩類Frank Copula模型參數(shù)的M-H抽樣算法；通過仿真分析給出指數(shù)Frank Copula模型參數(shù)的貝葉斯估計結(jié)果,利用貝葉斯p統(tǒng)計量檢驗估計的有效性和穩(wěn)健性,結(jié)果表明貝葉斯估計能準確估計參數(shù)。 然后,研究了基于刪失數(shù)據(jù)的Copula生存模型的貝葉斯推斷理論。包括異質(zhì)、正穩(wěn)態(tài)和治愈率刪失Copula生存模型構(gòu)建；推導(dǎo)異質(zhì)刪失Copula生存模型參數(shù)的條件后驗分布；設(shè)計Gibbs抽樣、自適宜和M-H抽樣算法對正穩(wěn)態(tài)刪失Copula生存模型邊際參數(shù)的估計；利用一步和兩階段貝葉斯估計分別推導(dǎo)相依參數(shù)的條件后驗分布；設(shè)計Gibbs抽樣推導(dǎo)治愈率刪失Copula生存模型參數(shù)的完全條件后驗分布。 利用刪失生存的實際數(shù)據(jù),分別用刪失正穩(wěn)態(tài)、Frank和Clayton Copula生存模型估計變量間相依結(jié)構(gòu),給出兩階段與一步貝葉斯估計的參數(shù)后驗統(tǒng)計量,然后利用DIC、EAIC、EBIC和CPO等統(tǒng)計量對所用模型進行比較選擇分析。 其次,研究了貝葉斯方法對邊際分布為連續(xù)、離散和混合變量的多元Copula模型參數(shù)估計和統(tǒng)計推斷理論。引入二元指示變量對相關(guān)矩陣參數(shù)化,設(shè)計M-H抽樣算法完成連續(xù)多元Copula模型的潛變量和參數(shù)化矩陣元素的估計。討論離散和混合變量的多元Copula模型構(gòu)建,利用MCMC抽樣得到邊際分布、潛變量和相依參數(shù)的條件后驗分布。構(gòu)建多元Copula回歸模型,討論協(xié)方差矩陣的先驗選擇,研究離散和混合變量情形下邊際分布參數(shù)和相關(guān)矩陣元素的MCMC抽樣過程。同時結(jié)合Monte Carlo仿真對混合變量的正態(tài)Copula模型的貝葉斯抽樣過程進行實現(xiàn),給出相關(guān)參數(shù)的后驗估計和檢驗。 最后,研究了基于時間序列的時變t-Copula模型的貝葉斯推斷理論。利用靜態(tài)Copula、時變Copula和時變Copula貝葉斯模型分別描述金融危機前后國際原油價格與亞太股票市場的相依結(jié)構(gòu)。研究結(jié)果表明,金融危機后相依結(jié)構(gòu)比危機前明顯增強,時變Copula模型更加適合刻畫變量間的相依結(jié)構(gòu),同時利用靜態(tài)Copula、時變Copula和時變Copula貝葉斯模型估計原油與亞太股票市場投資組合的VaR,發(fā)現(xiàn)時變t-Copula貝葉斯模型可以更好地估計投資組合的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 .

【學(xué)位授予單位】：湖南大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2014
【分類號】：F830.91;F224

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