基于三阶条件下的降偏差重尾指数估计
发布时间：2019-01-03 17:46
【摘要】：已有大量数据表明,金融、保险、网络、生物学以及风险理论等领域的时间序列数据并不满足正态分布,而是展现出尖峰厚尾的特征。为此,对重尾分布及尾指数估计的研究显得十分重要,如何对尾指数做出无偏且稳健的估计成为学者们关注的热点。本文首先介绍了极值理论和正则变化条件。其次,介绍了 MOP估计在一阶条件下的一致性与二阶条件下的渐近正态性,并且假设在较强的三阶条件下证明了MOP估计的渐近展开式,相比于较弱二阶条件下的结论得到关于渐近偏差的更多信息。然后,介绍了OMOP估计和ORBMOP估计,并在三阶条件下证明了 ORBMOP估计的渐近展开式和渐近正态性且在一定条件下获得了一个非零渐近偏差。在保证方差为γ2(1-φρ)2/(1-2φρ)的前提下,提出ORBMOP的一种修正估计(?)*(κ)并在三阶条件下证明了它的渐近正态性,当γ 0或γ 0时比较了二者的渐近偏差。最后,在有限样本情形下,利用三种常见重尾模型对本文提出的(?)*(κ)估计与ORBMOP估计和经典降偏差CH估计进行Monte-Carlo模拟比较,模拟均值表明(?)*(κ)估计比ORBMOP和CH更接近γ真值,模拟均方误差表明(?)*(κ)估计的均方误差更小。因此,本文提出的修正的降偏差估计(?)*(κ)表现更好。
[Abstract]:A large number of data have shown that the time series data in the fields of finance, insurance, network, biology and risk theory do not satisfy the normal distribution, but show the characteristics of peak and thick tail. Therefore, it is very important to study the heavy-tailed distribution and tail index estimation. How to estimate the tail index unbiased and robust has become a hot topic. In this paper, the extreme value theory and the regular variation conditions are introduced. Secondly, we introduce the consistency of MOP estimator under the first order condition and the asymptotic normality under the second order condition, and prove the asymptotic expansion of the MOP estimate under the strong third order condition. More information about the asymptotic deviation is obtained than that under the weaker second order condition. Then, the OMOP and ORBMOP estimators are introduced, and the asymptotic expansions and asymptotic normality of ORBMOP estimators are proved under the third-order conditions, and a nonzero asymptotic deviation is obtained under certain conditions. Under the condition that the variance is 纬 2 (1- 蠁 蟻) 2 / (1-2 蠁 蟻), a modified estimate of ORBMOP (?) * (魏) is proposed and its asymptotic normality is proved under the third-order condition. The asymptotic deviations are compared when 纬 0 or 纬 0. Finally, in the case of limited samples, three kinds of common heavy-tailed models are used to compare the (?) * (魏) estimators proposed in this paper with ORBMOP estimators and classical reduced deviation CH estimators. The simulated mean value shows that (?) * (魏) estimation is closer to 纬 true value than ORBMOP and CH, and the simulated mean square error indicates that the mean square error of (?) * (魏) estimation is smaller than that of ORBMOP and CH. Therefore, the revised reduced deviation estimate (?) * (魏) proposed in this paper performs better.
【学位授予单位】：山西大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.1
本文编号：2399695
[Abstract]:A large number of data have shown that the time series data in the fields of finance, insurance, network, biology and risk theory do not satisfy the normal distribution, but show the characteristics of peak and thick tail. Therefore, it is very important to study the heavy-tailed distribution and tail index estimation. How to estimate the tail index unbiased and robust has become a hot topic. In this paper, the extreme value theory and the regular variation conditions are introduced. Secondly, we introduce the consistency of MOP estimator under the first order condition and the asymptotic normality under the second order condition, and prove the asymptotic expansion of the MOP estimate under the strong third order condition. More information about the asymptotic deviation is obtained than that under the weaker second order condition. Then, the OMOP and ORBMOP estimators are introduced, and the asymptotic expansions and asymptotic normality of ORBMOP estimators are proved under the third-order conditions, and a nonzero asymptotic deviation is obtained under certain conditions. Under the condition that the variance is 纬 2 (1- 蠁 蟻) 2 / (1-2 蠁 蟻), a modified estimate of ORBMOP (?) * (魏) is proposed and its asymptotic normality is proved under the third-order condition. The asymptotic deviations are compared when 纬 0 or 纬 0. Finally, in the case of limited samples, three kinds of common heavy-tailed models are used to compare the (?) * (魏) estimators proposed in this paper with ORBMOP estimators and classical reduced deviation CH estimators. The simulated mean value shows that (?) * (魏) estimation is closer to 纬 true value than ORBMOP and CH, and the simulated mean square error indicates that the mean square error of (?) * (魏) estimation is smaller than that of ORBMOP and CH. Therefore, the revised reduced deviation estimate (?) * (魏) proposed in this paper performs better.
【学位授予单位】：山西大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.1
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