极值统计模型族的参数估计及其应用研究
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摘要
在人们的认知范围内,极值事件较少出现,然而一旦发生影响重大.极值统计理论的产生与发展,为这类随机事件的统计分析提供了理论依据.本文主要对极值统计模型族的特性、参数估计及其应用进行了研究.论文在深入研究极值统计理论的基础上,对极值统计模型族的适用范围进行了剖析;阐述了极值分布的Bayes参数估计方法;构建了基于极值分布理论和Copula函数的随机向量的相关模型,并对极值数据的尾部相关性进行了分析.论文的主要工作如下:
1.数理统计的核心内容是统计推断,其中参数估计是统计推断的主要内容之一.论文从Bayes参数估计入手,采用Markov Chain Monte Carlo(MCMC)方法,构建了极值分布的Bayes参数估计框架.在此基础上对黄浦江某水文观测站T年一遇的最高水位进行了估计.实例研究表明,用Bayes方法估计的最高水位稍高于用极大似然估计得到的结果.
2.相关性分析是金融资产风险投资的一个重要问题.论文把极值统计分布与Copula函数相结合,构建了相关模型—M-Copula-GPD模型.探讨了此模型的参数估计及假设检验问题,并运用此模型对上海、深圳两股票指数之间的相关结构进行了研究.结果表明,两市场之间是一种非对称的相关模式.
3.极值事件是随机小概率事件,位于概率分布图形的左右尾部区域,极值理论恰是着眼于对随机变量分布的尾部区域的研究.论文在讨论了两个尾部相关性度量指标,以及与其有关的尾部相关系数等尾部指标应用特点的基础上,对上海股票市场收益率与成交量之间的极值相关性进行了探讨.结果表明:收益率与成交量之间具有一定的相关性.
4.作为有效的金融风险度量工具,VaR已经被广泛接受,其计算方法也得到了不断的改进.论文对目前存在的几种计算VaR方法进行了分析和比较,提出了GARCH-GPD模型,并对深圳股市指数进行了实证研究.结果表明,GARCH-GPD模型能有效捕捉金融收益序列的尖峰厚尾、波动聚集等特性,在较高的置信水平下,GARCH-GPD模型显示的结果更加安全.进一步地,对CVaR进行了研究.
Extreme value events are rarer than those already recorded, but profoundin?uences are produced when they occur. The production and development ofextreme value theory provide theoretical bases for these random events. In thisdissertation, the properties and parametric estimations of extreme value modelsand their applications are studied intensively. As a result of these researches,several extreme value models based on extreme value theory are analyzed. TheBayes methodology to be used within extreme value analysis is also proposed.And dependence models based on extreme value theory and Copula function arealso investigated systematically. Some discussions are done for tail dependence ofextreme observations. The main achievements of this work are listed as follows:
1. The key point of the mathematics statistics is inference. And the para-metric estimation is one of the main contents in inference. In this paper aframework—the Bayes methodology is proposed and the Markov Chain MonteCarlo techniques are used to make random observations which have posteriordistribution. Annual data recorded of highest water level in survey stations ofHuangpu River are analyzed. And some results are slightly high compared withthose of the corresponding likelihood method.
2. Dependence analysis is a central issue in portfolio construction. Combinewith extreme value models and Copula functions together, M-Copula-GPD modelis established. Estimation and test methods of M-Copula-GPD are studied too.This model is used to study the degree and patterns of dependence betweenShanghai and Shenzhen stock markets. The empirical results show that non-symmetric pattern exists between the two markets.
3. Extreme events which locate at the two sides of the distribution aresmall rate a?airs from the meaning of statistics. And extreme value theory focuson these types of events by luck. Two dependence measures and associatedwith the coe?cient of tail dependence are provided in this paper. Based onthese discussions, Novel diagnostic measures for dependence about returns andtransaction volumes of Shanghai stock index are studied. The conclusions drawnare as follows: weak extremal dependence for the stock data.
4. As a valid financial risk tool, VaR has already been accepted extensively, and its calculation method also got a continuous improvement. Several methodsexisted currently to estimate VaR are analyzed and comprised. GARCH-GPDmodel are pointed and used to Shenzhen stock index. The empirical results showthat the GARCH-GPD model can thoroughly capture the volatility when fat-tailed densities are taken into account. And its conclusion is more secure thanother model’s. Moreover, CVaR is discussed in this paper.
1.数理统计的核心内容是统计推断,其中参数估计是统计推断的主要内容之一.论文从Bayes参数估计入手,采用Markov Chain Monte Carlo(MCMC)方法,构建了极值分布的Bayes参数估计框架.在此基础上对黄浦江某水文观测站T年一遇的最高水位进行了估计.实例研究表明,用Bayes方法估计的最高水位稍高于用极大似然估计得到的结果.
2.相关性分析是金融资产风险投资的一个重要问题.论文把极值统计分布与Copula函数相结合,构建了相关模型—M-Copula-GPD模型.探讨了此模型的参数估计及假设检验问题,并运用此模型对上海、深圳两股票指数之间的相关结构进行了研究.结果表明,两市场之间是一种非对称的相关模式.
3.极值事件是随机小概率事件,位于概率分布图形的左右尾部区域,极值理论恰是着眼于对随机变量分布的尾部区域的研究.论文在讨论了两个尾部相关性度量指标,以及与其有关的尾部相关系数等尾部指标应用特点的基础上,对上海股票市场收益率与成交量之间的极值相关性进行了探讨.结果表明:收益率与成交量之间具有一定的相关性.
4.作为有效的金融风险度量工具,VaR已经被广泛接受,其计算方法也得到了不断的改进.论文对目前存在的几种计算VaR方法进行了分析和比较,提出了GARCH-GPD模型,并对深圳股市指数进行了实证研究.结果表明,GARCH-GPD模型能有效捕捉金融收益序列的尖峰厚尾、波动聚集等特性,在较高的置信水平下,GARCH-GPD模型显示的结果更加安全.进一步地,对CVaR进行了研究.
Extreme value events are rarer than those already recorded, but profoundin?uences are produced when they occur. The production and development ofextreme value theory provide theoretical bases for these random events. In thisdissertation, the properties and parametric estimations of extreme value modelsand their applications are studied intensively. As a result of these researches,several extreme value models based on extreme value theory are analyzed. TheBayes methodology to be used within extreme value analysis is also proposed.And dependence models based on extreme value theory and Copula function arealso investigated systematically. Some discussions are done for tail dependence ofextreme observations. The main achievements of this work are listed as follows:
1. The key point of the mathematics statistics is inference. And the para-metric estimation is one of the main contents in inference. In this paper aframework—the Bayes methodology is proposed and the Markov Chain MonteCarlo techniques are used to make random observations which have posteriordistribution. Annual data recorded of highest water level in survey stations ofHuangpu River are analyzed. And some results are slightly high compared withthose of the corresponding likelihood method.
2. Dependence analysis is a central issue in portfolio construction. Combinewith extreme value models and Copula functions together, M-Copula-GPD modelis established. Estimation and test methods of M-Copula-GPD are studied too.This model is used to study the degree and patterns of dependence betweenShanghai and Shenzhen stock markets. The empirical results show that non-symmetric pattern exists between the two markets.
3. Extreme events which locate at the two sides of the distribution aresmall rate a?airs from the meaning of statistics. And extreme value theory focuson these types of events by luck. Two dependence measures and associatedwith the coe?cient of tail dependence are provided in this paper. Based onthese discussions, Novel diagnostic measures for dependence about returns andtransaction volumes of Shanghai stock index are studied. The conclusions drawnare as follows: weak extremal dependence for the stock data.
4. As a valid financial risk tool, VaR has already been accepted extensively, and its calculation method also got a continuous improvement. Several methodsexisted currently to estimate VaR are analyzed and comprised. GARCH-GPDmodel are pointed and used to Shenzhen stock index. The empirical results showthat the GARCH-GPD model can thoroughly capture the volatility when fat-tailed densities are taken into account. And its conclusion is more secure thanother model’s. Moreover, CVaR is discussed in this paper.
引文
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