测量误差模型方差变点的统计推断
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摘要
在统计学中,变点问题一直是一个比较热门的一个研究方向。一般认为变点问题的研究始于Page(1954)在Biometrika上发表的一篇关于连续抽样检验的文章。20世纪60年代后期,更多的统计学者投入到这一研究领域,发表了大量有关变点理论和应用的学术论文。特别是近三十年来,变点问题的研究在理论和应用等方面都有了快速的发展。其中Krisnaiah and Miao(1988),Csorgo and Horvath(1997)以及Chen and Gupta(2000),Wu(2005)是对变点研究领域近三十年来理论研究的总结。在应用方面,变点问题已不再限于早期在工业控制中的应用,目前在经济、金融、医学和计算机等方面都有大量的应用背景。
已有文献大多数是关于位置参数变点的研究,特别是关于均值变点的研究,而关于方差变点的研究相对比较少。本论文主要研究了当已知测量误差模型中误差方差至多存在一个变点时,应用不同方法给出了方差变点的估计以及变点估计量的相合性,收敛速度,同时进一步探讨了变点估计量在局部独立假设条件下的渐近分布,并与已有结果进行了模拟比较。
本论文共分为四章。第一章主要概述了变点问题的研究现状,从研究方法和研究内容上系统地概述了变点问题的研究进展和常见的三个研究方面。同时介绍了这几类变点常用的研究方法：参数方法和非参数方法,并着重介绍了U-统计量法在变点统计推断问题中的应用和方差变点的发展及检测方法。
在第二章中介绍了测量误差模型以及本文考虑的带有方差变点的测量误差模型：Wj=Xj+ζj其中：Xj是独立同分布(i.i.d)的不可直接观察的真实变量；Wj是变量Xj在有测量误差下的观测值；ζj是相互独立的0均值正态分布的测量误差,且ζj与Xj独立,j=1,...,n。当已知测量误差存在一个变点时,即：其中σ12≠σ22和k*均未知,本章利用随机变量的特征函数构造变点估计的统计量,并研究了估计量的强弱相合性和指数阶收敛速度,同时进行模拟分析。
第三章给出了测量误差模型方差变点的U统计量型估计量,即利用“U-统计量”的性质,在已知测量误差模型中至多存在一个误差方差的变点时,研究了方差变点估计量的强弱相合性以及收敛速度。并进一步地在局部对立假设条件下,即方差的变化幅度ηn=|σ22-σ12|→0(n→0)的条件下,研究了变点估计量(?)的渐近分布：nρn2(τn-τ*)→arg maxs V(s).其中：其中：λ1,λ2是与观察值wi的二阶距有关的量。
在第四章中,应用特征函数构造了估计方差变点的CUSUM型统计量,研究了变点估计量的强弱相合性以及收敛速度。并在局部对立假设条件下,进一步证明了变点估计量的渐近分布可表示为双边Brown运动。
Change-point is one of the main research field in statistics. It is known that Change Point Problems began in Page(1954) who published a thesis about the con-tinuous sampling in Biometrika. After the late1960s, more statisticians devote into this field of research, and a lot of theoretic and applied papers about change Point were published. Especially the last three decades, theory and applications of the Change Point Problems have a rapid development. Krisnaiah and Miao(1988), Csorgo and Horvath(1997), Chen and Gupta(2000), Wu(2005)are the theoretical research Sum-mary of the past three decades in this area. In the application, the change point problem is no longer limited in industrial control, it is widely applied in the economic, financial, medical and computer ect.
Most of literatures are focus on the study of location parameters change point, especially on the study of the mean change point, the literatures about the study of variance change point are relatively much less. In this thesis, when knowing that there is one variance change point in measurement error model, we make the estimator and study the consistency, convergence rate of the change point estimator in different meth-ods, we further study the asymptotic distribution of the estimator under the condition of the local alternative hypothesis, and make comparison with the existing results by simulation.
This thesis is divided into four chapters. In the first chapter, we give a brief overview of the change point problems, and elaborate the common three research ar-eas in change point problems, describes the parameters and non-parametric methods for dealing with the change point. Especially, in the first chapter we highlights the application of U-statistic method in the statistical inference for the change point and describe the detection method for variance change point in detail.
In chapter2, we first introduce the measurement error model and the measurement error model with a variance change point considered in this thesis. where{Xj} are the unable observed i.i.d variables, Wj are the contaminated versions of the independent variables Xj with measurement error ζj,ζj are independent normal variables with mean0and ζj are independent with Xj, j=1,..., n. when there is only one change point in the series of variances of{ζj}, i.e., where σ12≠σ22and ko is unknown. In this chapter, an estimate of a change point in variance of measurement errors (ME) is given in terms of characteristic functions when the variances are known. Its modification is also given for the case that the variances are unknown. In addition, the consistency and convergence rates of the estimator and its modification are investigated. The simulation study shows that the proposed estimators perform well.
In chapter3,we construct a U-statistics to estimate the variance change by char-acteristic functions. Taking the advantage of the nature of the "U-statistics ", the consistency and conver-gence rates of the estimator will be investigated assuming that there is only one change point in the series of variances of{ζj}-In addition, the asymptotic distribution of the change point estimate are researched under the condition of the local alternative hypothesis that is ηn=|σ22-σ12|→0(n→0). we have:nρn2(τn-τ*)→arg maxs V(s).where: λ1,λ2are depend on the second moment of the observation wi.
In chapter4, we construct a CUSUM type statistics to estimate the variance change by characteristic functions.The consistency and convergence rate of change point estimate are presented. In addition, we prove that the asymptotic distribution of the change point estimate can be expressed as a bilateral Brownian motion under the condition of the local alternative hypothesis.
已有文献大多数是关于位置参数变点的研究,特别是关于均值变点的研究,而关于方差变点的研究相对比较少。本论文主要研究了当已知测量误差模型中误差方差至多存在一个变点时,应用不同方法给出了方差变点的估计以及变点估计量的相合性,收敛速度,同时进一步探讨了变点估计量在局部独立假设条件下的渐近分布,并与已有结果进行了模拟比较。
本论文共分为四章。第一章主要概述了变点问题的研究现状,从研究方法和研究内容上系统地概述了变点问题的研究进展和常见的三个研究方面。同时介绍了这几类变点常用的研究方法：参数方法和非参数方法,并着重介绍了U-统计量法在变点统计推断问题中的应用和方差变点的发展及检测方法。
在第二章中介绍了测量误差模型以及本文考虑的带有方差变点的测量误差模型：Wj=Xj+ζj其中：Xj是独立同分布(i.i.d)的不可直接观察的真实变量；Wj是变量Xj在有测量误差下的观测值；ζj是相互独立的0均值正态分布的测量误差,且ζj与Xj独立,j=1,...,n。当已知测量误差存在一个变点时,即：其中σ12≠σ22和k*均未知,本章利用随机变量的特征函数构造变点估计的统计量,并研究了估计量的强弱相合性和指数阶收敛速度,同时进行模拟分析。
第三章给出了测量误差模型方差变点的U统计量型估计量,即利用“U-统计量”的性质,在已知测量误差模型中至多存在一个误差方差的变点时,研究了方差变点估计量的强弱相合性以及收敛速度。并进一步地在局部对立假设条件下,即方差的变化幅度ηn=|σ22-σ12|→0(n→0)的条件下,研究了变点估计量(?)的渐近分布：nρn2(τn-τ*)→arg maxs V(s).其中：其中：λ1,λ2是与观察值wi的二阶距有关的量。
在第四章中,应用特征函数构造了估计方差变点的CUSUM型统计量,研究了变点估计量的强弱相合性以及收敛速度。并在局部对立假设条件下,进一步证明了变点估计量的渐近分布可表示为双边Brown运动。
Change-point is one of the main research field in statistics. It is known that Change Point Problems began in Page(1954) who published a thesis about the con-tinuous sampling in Biometrika. After the late1960s, more statisticians devote into this field of research, and a lot of theoretic and applied papers about change Point were published. Especially the last three decades, theory and applications of the Change Point Problems have a rapid development. Krisnaiah and Miao(1988), Csorgo and Horvath(1997), Chen and Gupta(2000), Wu(2005)are the theoretical research Sum-mary of the past three decades in this area. In the application, the change point problem is no longer limited in industrial control, it is widely applied in the economic, financial, medical and computer ect.
Most of literatures are focus on the study of location parameters change point, especially on the study of the mean change point, the literatures about the study of variance change point are relatively much less. In this thesis, when knowing that there is one variance change point in measurement error model, we make the estimator and study the consistency, convergence rate of the change point estimator in different meth-ods, we further study the asymptotic distribution of the estimator under the condition of the local alternative hypothesis, and make comparison with the existing results by simulation.
This thesis is divided into four chapters. In the first chapter, we give a brief overview of the change point problems, and elaborate the common three research ar-eas in change point problems, describes the parameters and non-parametric methods for dealing with the change point. Especially, in the first chapter we highlights the application of U-statistic method in the statistical inference for the change point and describe the detection method for variance change point in detail.
In chapter2, we first introduce the measurement error model and the measurement error model with a variance change point considered in this thesis. where{Xj} are the unable observed i.i.d variables, Wj are the contaminated versions of the independent variables Xj with measurement error ζj,ζj are independent normal variables with mean0and ζj are independent with Xj, j=1,..., n. when there is only one change point in the series of variances of{ζj}, i.e., where σ12≠σ22and ko is unknown. In this chapter, an estimate of a change point in variance of measurement errors (ME) is given in terms of characteristic functions when the variances are known. Its modification is also given for the case that the variances are unknown. In addition, the consistency and convergence rates of the estimator and its modification are investigated. The simulation study shows that the proposed estimators perform well.
In chapter3,we construct a U-statistics to estimate the variance change by char-acteristic functions. Taking the advantage of the nature of the "U-statistics ", the consistency and conver-gence rates of the estimator will be investigated assuming that there is only one change point in the series of variances of{ζj}-In addition, the asymptotic distribution of the change point estimate are researched under the condition of the local alternative hypothesis that is ηn=|σ22-σ12|→0(n→0). we have:nρn2(τn-τ*)→arg maxs V(s).where: λ1,λ2are depend on the second moment of the observation wi.
In chapter4, we construct a CUSUM type statistics to estimate the variance change by characteristic functions.The consistency and convergence rate of change point estimate are presented. In addition, we prove that the asymptotic distribution of the change point estimate can be expressed as a bilateral Brownian motion under the condition of the local alternative hypothesis.
引文
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