【作者】 谭常春；

【导师】 缪柏其；

【作者基本信息】 中国科学技术大学 ， 概率论与数理统计， 2007， 博士

【摘要】 变点问题自上世纪70年代以来一直是统计中的一个热门话题，目前它不但在工业质量控制（最早产生变点问题统计研究的领域之一）中有广泛的应用，而且在经济、金融、医学、计算机等领域也还有大量的应用和应用背景。本论文从一个侧面用大样本理论集中探讨了г-分布变点估计和检验的统计理论和具体应用。本文前四章探讨了变点的统计理论研究，最后一章介绍了变点理论在金融中的应用。首先我们在第一章简要概述了变点问题的发展和研究现状。由于以往的研究主要集中于均值和方差变点的研究，本质上这两个参数是独立变化的，但是在г-分布中，均值和方差同时依赖于两个参数ν，λ。如何对这类分布参数的变点进行检测是人们关心的一个问题。由于有一定的难度，这类问题在文献中讨论得较少，我们选择了很有代表性的г-分布簇进行这方面的研究。对至多一个变点的г分布，即X₁，…，X_n为一列相互独立的随机变量序列，且X₂，…，X_[nτ₀] i．i．d～г（x；ν₁，λ₁），X_[nτ0]+1，…，X_n i．i．d～г（x；ν_z，λ₂），其中τ₀未知，称τ₀为该序列的变点。借助Gauss过程理论，利用第一型极值分布逼近文中提出的统计量的分布。在第二章，我们采用局部比较法讨论了г-分布参数变点τ₀的假设检验问题、变点估计（?）的强弱相合性以及收敛速度，并且比较了弱相合和强相合收敛的速度，最后给出了Matlab模拟。在第三章中我们利用CUSUM方法讨论了г-分布参数是否存在变点的假设检验问题、检测变点τ₀位置的程序、变点的估计（?）的强弱相合性和强弱收敛速度。同时对变点处（假定已检测有变点）的变异系数ν^1/2和跳跃度ν_n=ν₂^1/2-ν₁^1/2进行估计，并给出了估计量的渐近分布。当跳跃度ν_n=ν₂^1/2-ν₁^1/2较小时，给出了变点估计（?）的渐近分布。在方差未知时，用自正则方法给出了变点的检测方法，并给出了Matlab模拟结果。对至多一个变点的模型X（i/n）=f（i/n）+ε（i/n），其中f（t）=，ε（1/n），ε（2/n），…，ε（n/n）独立同分布。借助Gauss过程理论、第一型极值分布理论和局部比较法，在第四章我们不仅证明了变点丁τ₀估计（?）的强弱相合性，并给出了强弱收敛速度。进一步讨论了局部对立假设下，（?）的渐近分布。最后，我们给出了г-分布变点统计理论的一个应用，即讨论了上海股市股票指数连涨和连跌收益率分布中的变点问题。实证结果表明，在实行“T+0”、“T+1”、“涨停板”、“国有股减持”时期连涨收益率、连跌收益率虽然都服从г-分布，但是不同时期г-分布的形状参数和刻度参数却不相同，反映了中国股市发展过程中呈现出“政策市”的特点。更多还原

【Abstract】 Change-point problems have originally arisen in the context of quality control, where one typically observes the output of a production line and would wish to signal deviation from an acceptable level while observing the data. It also occurs in many natural and social fields such as economic, finance, seismology, epidemiology and computer science. In this thesis, we discuss mainly the statistical inference of change-point problems inΓ-distribution and its application in finance.This thesis is divided into two parts. In the first part, including the first four chapters, the statistical inference of change-point problems are mainly considered. To apply the theory and methods of change-point problems in practice, we discuss the application of change-point problems in finace in the second part （Chapter five）.The development and research status of change-point problems are briefly introduced in the first chapter.Former research focused on change-point problems in mean and variance mostly, where the mean and variance are independent parameters. But the mean and variance ofΓ-distribution depended on common parameters v and A simultaneously. It is an interested topic that how to detect the change-point in distribution with multi-parameter which are interrelated. In the second chapter, the change-point of parameters inΓ-distribution is considered. Suppose that are independent random variables, such that is unknown and called change-point. With help of the theory of Gaussian process and slipping window, the asymptotic distribution of the statistic proposed in this chapter can be approximated by the first type of extremal distribution. Hypothesis test and estimation about change-point are considered. Furthermore, the consistency and convergence rate of change-point are also presented. At the same time, strong rate and weak rate of convegence of change-point are established. Lastly, simulations by Matlab are displayed, it is shown that our procedures are rather accurate.In chapter three, the procedures about detectection and estimation of change-point are proposed with the help of CUSUM. In the mild conditions, the strong consistency and convergence rate of estimator of change-point are also presented. The estimation of the magnitude of jump v^1/2 = v₂- v₁ and the asymptotic distribution of estimator are also discussed. At the same time, the asympotic distribution of estimator of change point is also presented when the variance of X_i is known. The procedures to detect the change-point are also proposed by mean of self-normalization when the variance of X_i is unknown. At last, the results of simulation is presented, it is shown that our test and estimator are rather efficient.In the fourth chapter, we discussed the change-point problem with at most one change inthe model: where are independent and identically distributed. The strong and weak consistency of estimator of change-point are considered. The strong and weak convergence rate of estimatorτ|^ of change-point are also given. In the mild conditions, the asymptotic distribution of estimatorτ|^ is presented.As a practical application of change-point problem, in the fifth chapter we research the successive rises and falls of Shanghai stock returns by change-point analysis methods inΓ-distribution. Based on the the example of Shanghai stock index returns, the following result is gained: The distributions of the successive rises and falls of returns are fitted with Gamma- distribution well in period " T+0 ", " T+1" " rise and stop " " reduce state stake in listed companies " respectively, but the shape parameter and scale parameter inΓ-distribution are not the same in different periods. This reflects the trait of " policy " on the course of development in Chinese stock market.更多还原