Extreme Value Statistics And Quantile Regression: Theory And Application

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.