一类广义相依序列的概率不等式及统计推断研究

Acceptable random variables and Extended negatively dependent random variables and Wide dependent random variables is a kind of random variables as generalization of Positive and Negative dependent random variables. So far, the study of probability inequality is very less for this kinds of random variables. That leads to the lack of methods and tools of studying its related statistical inference. Therefore the major issues of this project focus on the following questions: i) Based on the kind of dependent sequence, to establish several important probability inequalities as Rosenthal type inequality, moment inequalities, Bernstein type inequality, and so on. ii) By these inequalities, under full date and censored data, the asymptotic properties of the weighted kernel estimators and wavelet estimators for semiparametric regression model will be discussed for these dependent random variables mentioned above, and to establish also the large sample properties for the estimators of unknown density function and distribution function. iii) And to further study the above statistical inference problem by using empirical likelihood method. iv) In view of this kind of dependent random variables widely applied in the risk analysis, and in order to extend the range and practicability for the methods of risk measurement, by using nonparametric method and empirical likelihood method, we will further investigate the asymptotic properties of the estimators for risk measurement as Expected Shortfall, Value at Risk and Conditional Expected Shortfall. This project research can further perfect and develop the probibality inequality and statistical inference theory of dependent variable.

可接受随、广义负相依和宽相依随机序列是正负相协等随机序列的一般化和推广。迄今，对这类广义相依序列概率不等式的研究较少，导致对相关统计模型的统计推断问题的研究缺乏必要的工具。为此，本项目将主要研究：1)基于这类广义相依序列， 建立Rosenthal不等式、矩不等式和Bernstein型等概率不等式；2)利用这类广义相依序列概率不等式，在完全数据和删失数据下，讨论半参数回归模型中未知参数和函数的加权核估计和小波估计的渐近性质，以及总体密度和分布函数递归核估计的大样本性质；3)利用经验似然方法等, 继续深入研究上述模型的统计推断问题；4)鉴于这类广义相依序列在风险分析中有广泛应用，为了拓广风险度量方法的适应范围和实用性，将利用非参数方法和经验似然方法，进一步研究期望损失、风险价值和条件期望损失等风险度量的估计量的渐近性质。通过本项目研究，将进一步完善相依变量的概率不等式理论及其统计推断理论。

广义负相依、宽相依和负超可加相依随机序列是正负相协等随机序列的一般化和推广, 但研究统计模型估计的大样本性质比较少。本项目我们主要研究了：1)基于广义相依序列， 建立Rosenthal不等式、矩不等式和Bernstein型等概率不等式；2)利用广义相依序列概率不等式，在完全数据下，讨论半参数回归模型中未知参数和函数的加权核估计和小波估计的渐近性质，以及总体密度和分布函数递归核估计的大样本性质；3)在删失数据下，讨论了生存函数的Kaplan-Meier估计和风险率函数估计量的强逼近和强表示式等大样本性质；4)利用经验似然方法研究了严平稳混合样本密度函数高阶导数、EB检验统计量渐近最优性及其收敛速度等统计推断问题；5) 利用非参数方法，研究了VaR估计量的Bahadur表示、CVaR (conditional value at risk)的Berry-Esseen型界等统计大样本性质。项目所得研究结果进一步完善量相依变量的概率不等式理论及其若干统计模型中的统计推断理论，有较好的理论推广价值和实际应用前景。