【作者】 韩月丽；

【导师】 史道济；

【作者基本信息】 天津大学 ， 管理科学与工程， 2009， 博士

【摘要】 极值统计是专门研究很少发生,然而一旦发生却有巨大影响的随机变量极端变异性的建模及统计分析方法。分位数回归是给定回归变量X,估计响应变量Y条件分位数的一个基本方法,不仅可以度量回归变量在分布中心的影响,而且还可以度量在分布上尾和下尾的影响。本文主要对极值的基本理论、复合极值分布参数的估计方法、风险价值VaR的方差、分位数回归的理论、Copula分位数回归以及极值统计模型和分位数回归在各个领域的应用进行了深入研究。论文的主要工作如下:1.论文介绍了极值的基本理论,并应用二元超阈值模型和二元点过程模型讨论食物支出与家庭收入的相关性。结果表明:食物支出与家庭收入具有较强的相关性,而且这两种模型都不失为一种很好的建模方法。2.论文基于海洋工程中提出的复合极值分布,将变量赋予新的金融含义,构建金融风险管理领域的Poisson-Gumbel复合极值分布模型,提出采用概率权矩法进行参数估计,给出具体的计算结果,并通过蒙特卡洛模拟方法对概率权矩法和复合矩法进行了比较研究。结果表明:概率权矩法比复合矩法估计效果好而且表现稳定。3.论文研究了风险价值VaR的方差,并给出Poisson-Gumbel复合极值分布模型VaR的方差和Poisson-GP复合超阈值分布模型VaR的方差。在此基础上,用两个模型拟合1990年1月2日至2006年12月29日期间美元/英镑的汇率数据进行实证分析。4.论文构建线性条件分位数回归模型,分析房屋贷款与家庭收入之间条件分位数的线性变化趋势,并与经典的最小二乘回归拟合进行比较。结果表明:在不同分位数下房屋贷款与家庭收入之间所呈现的线性趋势是不同的,分位数回归比经典的最小二乘回归能够提供更多的信息。5.论文介绍了Copula分位数回归,并推导出几种常见Copula的分位数曲线。在此基础上,通过对Frank Copula进行模拟研究,显示了分位数回归估计方法的精确性。更多还原

【Abstract】 Extreme value theory is a branch of statistics dealing with the extreme deviation from median of probability distributions for highly unusual events, which will result an enormous impact on random variables when happening. Quantile regression is a statistic method that estimates the conditional quantiles of a response variable Y given X=x. It can be used to measure the influence not only on center but also on upper and lower tail of a probability distribution of an independent variable. This dissertation studied in depth extreme value theory, parameter estimation method of compound extreme value distribution, variance of Value at Risk (VaR), quantile regression theory, Copula quantile regression, and applications of extreme value statistics model and quantile regression in various fields. The major parts of the dissertation are listed below.1. Extreme value theory is introduced in the dissertation. The relationship between food expense and household income is discussed by using bivariate peaks over threshold model and bivariate point process model of extreme value theory. The results indicate that there exists stronger relationship between food expense and household income. Both models can be good approaches to similar problems.2. Based on the compound extreme value distribution proposed in ocean engineering, a Poisson-Gumbel compound extreme value distribution model for the field of financial risk management is built by given variables new meaning in the dissertation. It is further proposed to estimate parameter using probability-weighted moment method. The estimated results using Monte Carlo simulation based on probability-weighted moment method and complex moment method are compared. By comparison, the probability-weighted moment method is superior than complex moment method in terms of accuracy and robustness of estimations.3. The dissertation studied variance of VaR models and determined the variance of VaR for the Poisson-Gumbel compound extreme value distribution and the variance of VaR for the Poisson-GP compound peaks over threshold distribution. Case study is conducted using data of exchange rates between US Dollars and British Pounds from January 2, 1990 to December 29, 2006.4. A linear conditional quantile regression model is proposed in the dissertation. The relationship between mortgage payment and household income for both Chinese households and American households is analyzed and compared based on linear conditional quantile regression as well as ordinary linear regression. The results show that the relationships between mortgage payment and household income are different for differentτvalues. Compared to ordinary linear regression, quantile regression can reveal more regional information.5. Copula quantile regression is introduced in the dissertation. Several common Copula quantile curves are derived. Simulation study is performed based on Frank Copula. The results indicate that estimations using quantile regress are more accurate.更多还原