The Bayes Inference Of Several Kinds Of Distribution And The Extreme Value Theory For VaR Calculation

The classical statistics and Bayesian statistics are two schools in the mathematical statistics. Their biggest difference lies in how to understand the parameters of distribution.The Bayes theory considers the unknown parameters as random variables. It can always get a good result using the small samples with the considerations of a priori information.Firstly, under the conjugate prior distribution, Bayes estimates and conservative conditions of the loss and risk functions of the parameters in Normal, Lognormal, Rayleigh and Frechet distributions are given, and the rationalities of these conditions are discussed in the thesis. Then the data respectively on weekly closed prices of China Petrochemical and Ordos stocks in CSI 300, average wages of employees in different industries in cities and towns of China(2009) and daily returns of the SSE Composite Index are analyzed to support the finding in the thesis. Secondly, the Bayes estimate of a parameter with its admissibility problem is a hotspot in the field of statistics. There are many asymmetric risk situations caused by overestimate or underestimate in practice, so Bayesian estimate and its admissibility for the scale parameter of inverse gamma distribution and the shape parameter of generalized exponential distribution are discussed under a modified linear exponential loss function named Mlinex loss function, and also for the admissibility of the inverse linear form of a sufficient statistics. Then Monte Carlo simulation is used to clarify the better performance of Bayes estimate than maximum likelihood and the minimax estimates, and which is as good as the uniformly minimum variance unbiased estimate from estimated error for small samples.In addition to the above research, risk measurement problem in financial market is also explored in the thesis. As the global economic integration and increasing volatility in financial markets, VaR has become the international mainstream measurement of risk.Financial extreme events happen frequently in recent years and tremendous extreme risk in financial markets is widely discussed. The common VaR measurement methods are difficult to meet the study of extreme risk. The extreme value theory is about extreme value distribution characteristics of ordinal statistic without considering the particular form of return distribution. It can fit and measure the risk of loss data with thick tails better.Using ZTE daily closed prices data with heavy tails as an example, the BMM and the POT model of the extreme value theory are employed to fit the loss distribution with long positions and compute the VaR and ES under the fitting model respectively. A discrepancymeasure is proposed to select the threshold for the POT model, which is one of the innovation points in the thesis. It can avoid the subjective randomness of conventional methods. The study shows that comparing with the BMM, the POT model can better capture the extreme data information and obtain more realistic results of risk measurement.The result can provide a reference for investors to control risk.