贝叶斯框架下风险度量的非参数估计及其应用研究

The most important problems for management decision-makers in finance, bank and insurance, are to measure, properly estimate, and lastly to control risks in some way. This project research mainly the nonparametric estimate of risk measure and their statistical inferences based on Bayes theory though combination of sample information and prior information. From the references available in the fields of risk measure estimate, there are mainly two categories, one is based on large sample theory of classical statistics, another is based on parametric Bayesian theory. However, these results inevitably assume some specify distribution for risk variable and the prior parameters. The project will develop the distribution-free estimate from the following way: (1) Bayesian nonparametric methods. We use Furguson's Bayesian nonparametric method to estimate the risk measure, which can estimate risk measure without specific risk distribution; (2) The constrained Bayesian and empirical Bayesian method. The estimation of risk measures are constrained to a certain class of sample functions under minimum mean square error, and then combined with empirical Bayes method to estimate the unknown structure parameters. In this way, the estimator derived do not specify not only risk distribution but also prior distribution of prior parameters. (3) The combination of empirical Bayes and Bayesian nonparametric methods. We combine empirical Bayes and Bayesian nonparametric methods to get rid of the assumptions for risk distribution.

如何更好地度量、评估并控制风险是金融、银行、保险等管理者和决策者最为关心的问题。本课题拟基于贝叶斯理论，结合样本信息和先验信息，研究风险度量的非参数估计及其统计推断。在目前大多数风险度量的估计相关文献中，主要分为两类，一类是基于统计大样本理论，另一类是基于参数贝叶斯原理。但是，这些结果都不可避免对风险的分布或参数的先验分布有所假设。本项目从以下几点去寻找风险度量的真正无分布非参数估计：（1）非参数贝叶斯方法，利用Furguson的非参数贝叶斯方法估计风险度量，这时不需要假设风险具体的样本分布；（2）限制贝叶斯和经验贝叶斯方法，将所估计的风险度量限制在某些特定的函数类中，得到均方误差最小的估计，再利用经验贝叶斯方法估计未知超参数，此时不仅不需要风险的分布假设，而且不需要参数的先验分布假设；（3）经验非参数贝叶斯方法，将经验贝叶斯方法和非参数贝叶斯方法相结合，去掉传统的分布假设。

本项目从以下几点研究了风险度量的估计：（1）非参数贝叶斯方法，利用非参数贝叶斯方法估计风险度量，这时不需要假设风险具体的样本分布；（2）限制贝叶斯和经验贝叶斯方法，将所估计的风险测度限制在某些特定的函数类中，得到均方误差最小的估计，再利用经验贝叶斯方法估计未知的超参数，此时不仅不需要风险的分布假设，而且不需要参数的先验分布的假设；（3）经验非参数贝叶斯方法，将经验贝叶斯方法和非参数贝叶斯方法相互结合，去掉传统的风险分布假设。本项目的研究得到了很多有意义的成果，能为风险管理部门提供参考。