两类半参数统计模型中的估计方法
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摘要
变系数模型(Varying coefficient model)(也称为函数系数模型Functional coefficient model)是一般线性模型的一种有用推广。由于变系数模型能够涵盖许多常见的模型，因而引起了许多统计工作者的研究兴趣。变系数模型既部分保留了非参数回归的特点，又具有结构简单、模型容易解释等优点，因而在生物、医学医药研究、经济金融等领域的统计分析中得到了广泛的应用。
本文尝试用小波方法对变系数模型进行研究。小波作为一种技术，最初主要用于地震数据分析，后来随着其理论的发展，在图象处理、信号分析、数据压缩等方面也有了广泛的应用。但小波应用于统计还是上世纪九十年代初的事，由于小波理论为数理统计工作者提供了强有力的新技术，具有种种优良性，因而在统计领域得到了广泛的重视与应用，尤其近几年小波在统计学的应用方面的有关研究有了飞速的发展。
本文首先研究了Hastie和Tibshirani(1993)提出的变系数模型，运用小波方法建立了未知系数函数以及模型误差方差的估计量，并讨论了这些估计量的渐近性质。本文还讨论了另一类非参数估计问题，课题来源于复旦—瑞士再保险研究基金和国家自然科学基金项目，我们的研究结果为老年互助保险项目的研究提供了一种切实可行的方法。本课题旨在注重理论和实践相结合，体现学科的交叉与融合。以下是各章的主要内容介绍：
本文第一章简略回顾了统计模型的发展与演变，主要介绍了变系数模型的几种不同形式以及相关的各种估计方法，并列出了已有的一些研究成果，同时指出了变系数模型在实践中的可行性与广泛性。同其它方法相比，小波方法有许多优点，比如相对于正交级数估计，它对待估函数的要求较低，而得到的大样本性质较为理想。小波还具有能十分精确地描述出复杂曲线局部特征的能力，因而受到许多工程师、数学及统计工作者的广泛关注。因此本论文的第二、第三章将使用小波估计方法，与此相关在本章简略介绍一些小波的有关知识。在这一章的最后阐述了论文的主要研究内容和结果，并作了小结。
本文的第二章讨论如下形式的变系数模型：y=x_1β_1(t)+…+x_pβ_p(t)+e其中y为响应变量，x=(x_1，…，x_p)和t为协变量，e为随机误差，且E(e)=0，E(e~2)=σ~2。{β_j(t)，j=1，…，p}为未知的非参数函数，不失一般性，假设β_j(t)的定义域为[0，1]。σ~2是未知参数。该模型可以看作是一般线性模型的推广，它允许回归系数为某一回归变量t的函数，β_j(·)能够比较详细地刻画出x_j与t的相互关系。另外由于β_j(·)为一般的非参数函数，回归模型的偏差显著地减少，有力地避免了“维数祸根”现象。本文假设回归变量x是随机设计，而t是固定设计的情况下，将非参数回归小波的估计方法推广到上述变系数模型，建立了函数系数{β_j(t)，j=1，…，p}的小波估计并在较弱的条件下得到了估计的强一致收敛性和渐近正态性。
本文第三章讨论了上述变系数模型误差方差的估计问题，提出了误差方差σ~2的小波估计(?)_n~2，并证明了它的大样本性质，同时还构造了var(e~2)的小波估计(?)_n~2，证明了(?)_n~2的弱相合性，并由此得出n~(1/2)((?)_n~2-σ~2)/(?)_n(?)N(0，1)。这一结果可用于构造σ~2的大样本区间估计或对σ~2进行大样本检验。在相同的模型下，与已有的研究结果相比较，我们对待估函数的要求较低，而得到的估计性质较为理想，并且研究的结果可用于统计目的，这是有实际意义的。
本文第四章讨论了老年互助保险项目研究中生存函数的估计问题。课题的背景源于保险精算，在老年互助保险中，需要估计健康寿命(能自理)、半自理状态寿命及完全不能自理状态寿命的生存函数，能够知道的信息是只有在60岁以后的各个状态的样本，而要估计的是在60岁以前的某个有限生命区间内各状态的生存函数。其特殊性在于样本取值区间与生存函数的考察区间并不一致，因此通常的经验分布方法并不完全适用。
设X_1，X_2，…，X_n是来自F(x)的独立样本，当F(x)的泛函形式未知，但可用[0，1]上的多项式θ_1X±…±θ_rX~r逼近时，郑祖康(1995)已提出一种混合矩估计方法，并建立了估计量的强相合性，只是郑祖康(1995)的方法假定分布函数的支撑集为[0，1]，限制了它的应用范围。本文讨论F(x)的泛函形式未知，但经过logit变换后可用多项式逼近的情形，这就去掉了关于支撑集为[0，1]的限制。设F(x)有密度函数f(x)，且存在某个r≥1，(r未知)，使得logF(x)/(1-F(x))=a_0+a_1x+…+a_rx~r(?)J_r(x)其中x∈(a，b)，(a，b)(?)(a_0，b_0)。由最小二乘估计的思想，我们得到了逼近多项式的系数{a_0，a_1，…，a_r)的估计，进一步得到了分布函数F(x)的估计量，并且证明了系数估计{(?)_0，(?)_1，…，(?)_r}的一致收敛性和分布函数估计(?)(x)的一致收敛性。我们在主要讨论分布函数F(x)的同时，也得到了其密度函数f(x)和失效函数λ(x)的非参数估计及其相应的大样本结果。模拟结果显示估计是理想的。
本文最后作了总结与展望。
Varying coefficient model, also known as functional coefficient model, is auseful extension of the general linear model. Since varying coefficient model cancover a lot of other models, it has attracted the interest of many statistical researchers.Varying coefficient models not only retains partially the characteristics ofnonparametric regression, but is also simple in structure and easy to explain, whichhas led to its extensive application in statistical analysis in biology, medicine andmedicinal research, economics and finance.
This paper conducts research on varying coefficient model via wavelet method.As a technique, wavelet method was first used in the analysis of seismic data. With itstheoretical development, the method has been also found extensive application inimage processing, signal processing and data compressing. The early 1990s haswitnessed the application of wavelet method in statistical studies. Since waveletmethod provides a powerful new technique for statistical researchers and hasdemonstrated its various advantages, it has been highly valued, and recent years, it hasbeen rapid increase in the use of wavelet method in applied statistical studies.
This paper first studies the varying coefficient model by Hastie and Tibshirani(1993), and establishes unknown function coefficients and variance error estimator, and discusses asymptotic properties of estimators. The paper also discusses anothertype of nonparametric estimation. This research was financed by Fudan-SwitzerlandReinsurance Fund and the National Natural Science Foundation of China, and ourresearch findings have provided a practical method for research in mutual insurancefor the elderly. This paper tries to combine theory with practice, and realizes the crossconnection of different disciplines. The major contents of each chapter are as follows:
Chapter One summarizes the development and evolution of statistical models, withemphasis on variations of varying coefficient models and related estimation methods.Research achievements are outlined, and feasibility and extensiveness of the application of varying coefficient model are documented. Compared with othermethods, wavelet method has many advantages. For example, compared withorthogonal series estimation, wavelet method has a low demand on function, while thelarge sample resulting is more satisfactory. Wavelet method can accurately describethe local features of complex curves, which has caught the attention of manyengineers and researchers in mathematics and statistics. This chapter will give ageneral instruction to wavelet method so as to lay a basis for Chapters Two and Threein which wavelet method will be employed in analysis. At the end of this chapter, major research contents and results are elaborated, and a summary is given.
Chapter Two discusses the following varying coefficient model: y=x_1β_1(t)+…+x_pβ_p(t)+ewhere y is response variable, x=(x_1,…, x_p)~T and t are covariables, e is randomerror, whereas E(e)=0, E(e~2)=σ~2. {β-j(t), j=1,…, p} are unknownnonparametric functions, without losing generality, let the domain ofβ_j(t) be [0, 1].σ~2 is unknown parameter. This model can be seen as an extension of general linearmodel, which allows regression coefficient to be the function of a regression variablet.β_j(·) is able to describe in detail the relationship between x_j and t. In addition, sinceβ_j(·) is a general nonparametrie function, the error of regression model isnoticeably reduced, hence strongly avoid the phenomenon of "curse of dimension". Inthis paper, let x be random design, while t be fixed design. Apply nonparametricregression wavelet estimation to the afore-mentioned varying coefficient model, wecan establish wavelet estimation of function coefficients {β_j(t), j=1,…, p}, and canget strong uniform convergence and asymptotic normality.
Chapter Three discusses with error variance estimation of varying coefficientmodel mentioned previously, proposes wavelet estimation (?)_n~2 for error varianceσ~2, and proves its large sample nature. It also establishes wavelet estimation (?)_n~2 for var(e~2), and proves the weak consistence of (?)_n~2, from whichn~(1/2)((?)_n~2-σ~2)/(?)_n(?)N(0, 1) is deduced. This result can be used to construct largesample interval estimation or conduct large sample test forσ~2. For same model, compared with existing research results, this method has a weak demand for functionsto be estimated while achieving more satisfactory estimation property. In addition, theresearch results here can be used for statistical purposes, so it has practicalapplications.
Chapter Four discusses the estimation of survival functions in "mutual insurancefor the elderly project". The project stems from intensive computation of insurance.In mutual insurance for the elderly, we need to conduct estimation for survivalfunctions of self-sustaining life span, semi self-sustaining life span andnon-self-sustaining life span. The only know information is the sample of individualstate after the age of 60. What needs to be estimated is the survival functions of agiven life period prior to 60. What is special is that the sample-taking interval isdifferent from that for survival function, therefore traditional empirical distribution isnot totally applicable.
Let X_1, X_2,…, X_n be independent sample from F(x), when functional form ofF(x) is unknown but it can be approximated with polynomial on [0, 1]θ_1X±…±θ_rX~r, Zheng Zukang (1995) proposed the mixed moment estimation andestablished strong consistence of estimators. But Zheng Zukang's (1995) approachassumes that the support set for distribution function is on [0, 1], this has limited itsscope of application. This paper discusses unknown function form F(x) which canbe approximated after logit conversion, thereby eliminates the limit of support set[0, 1]. Let F(x) has density function f(x), and there exists a certain r≥1, (r isunknown) which satisfies log F(x)/1-F(x)=a_0+a_1x+…+a_rx~r(?)J_r(x)where x∈(a, b), (a, b)(?)(a_0, b_0). According to least square estimation theory, wecan get the estimation of approximating polynomial coefficients {a_0, a_1,…, a_r}, onwhich we get the estimators for distribution function F(x), and prove the uniformconvergence of coefficient estimation {(?)_0, (?)_1,…, (?)_r} and distribution function (?)(x).While discussing distribution function F(x), we also get the nonparametricestimation and corresponding large sample results for its density function f(x)andfailure functionλ(x). The simulation results indicate that estimation is satisfactory.
Last chapter outlines research achievements and assumes related studies in thefuture.
本文尝试用小波方法对变系数模型进行研究。小波作为一种技术，最初主要用于地震数据分析，后来随着其理论的发展，在图象处理、信号分析、数据压缩等方面也有了广泛的应用。但小波应用于统计还是上世纪九十年代初的事，由于小波理论为数理统计工作者提供了强有力的新技术，具有种种优良性，因而在统计领域得到了广泛的重视与应用，尤其近几年小波在统计学的应用方面的有关研究有了飞速的发展。
本文首先研究了Hastie和Tibshirani(1993)提出的变系数模型，运用小波方法建立了未知系数函数以及模型误差方差的估计量，并讨论了这些估计量的渐近性质。本文还讨论了另一类非参数估计问题，课题来源于复旦—瑞士再保险研究基金和国家自然科学基金项目，我们的研究结果为老年互助保险项目的研究提供了一种切实可行的方法。本课题旨在注重理论和实践相结合，体现学科的交叉与融合。以下是各章的主要内容介绍：
本文第一章简略回顾了统计模型的发展与演变，主要介绍了变系数模型的几种不同形式以及相关的各种估计方法，并列出了已有的一些研究成果，同时指出了变系数模型在实践中的可行性与广泛性。同其它方法相比，小波方法有许多优点，比如相对于正交级数估计，它对待估函数的要求较低，而得到的大样本性质较为理想。小波还具有能十分精确地描述出复杂曲线局部特征的能力，因而受到许多工程师、数学及统计工作者的广泛关注。因此本论文的第二、第三章将使用小波估计方法，与此相关在本章简略介绍一些小波的有关知识。在这一章的最后阐述了论文的主要研究内容和结果，并作了小结。
本文的第二章讨论如下形式的变系数模型：y=x_1β_1(t)+…+x_pβ_p(t)+e其中y为响应变量，x=(x_1，…，x_p)和t为协变量，e为随机误差，且E(e)=0，E(e~2)=σ~2。{β_j(t)，j=1，…，p}为未知的非参数函数，不失一般性，假设β_j(t)的定义域为[0，1]。σ~2是未知参数。该模型可以看作是一般线性模型的推广，它允许回归系数为某一回归变量t的函数，β_j(·)能够比较详细地刻画出x_j与t的相互关系。另外由于β_j(·)为一般的非参数函数，回归模型的偏差显著地减少，有力地避免了“维数祸根”现象。本文假设回归变量x是随机设计，而t是固定设计的情况下，将非参数回归小波的估计方法推广到上述变系数模型，建立了函数系数{β_j(t)，j=1，…，p}的小波估计并在较弱的条件下得到了估计的强一致收敛性和渐近正态性。
本文第三章讨论了上述变系数模型误差方差的估计问题，提出了误差方差σ~2的小波估计(?)_n~2，并证明了它的大样本性质，同时还构造了var(e~2)的小波估计(?)_n~2，证明了(?)_n~2的弱相合性，并由此得出n~(1/2)((?)_n~2-σ~2)/(?)_n(?)N(0，1)。这一结果可用于构造σ~2的大样本区间估计或对σ~2进行大样本检验。在相同的模型下，与已有的研究结果相比较，我们对待估函数的要求较低，而得到的估计性质较为理想，并且研究的结果可用于统计目的，这是有实际意义的。
本文第四章讨论了老年互助保险项目研究中生存函数的估计问题。课题的背景源于保险精算，在老年互助保险中，需要估计健康寿命(能自理)、半自理状态寿命及完全不能自理状态寿命的生存函数，能够知道的信息是只有在60岁以后的各个状态的样本，而要估计的是在60岁以前的某个有限生命区间内各状态的生存函数。其特殊性在于样本取值区间与生存函数的考察区间并不一致，因此通常的经验分布方法并不完全适用。
设X_1，X_2，…，X_n是来自F(x)的独立样本，当F(x)的泛函形式未知，但可用[0，1]上的多项式θ_1X±…±θ_rX~r逼近时，郑祖康(1995)已提出一种混合矩估计方法，并建立了估计量的强相合性，只是郑祖康(1995)的方法假定分布函数的支撑集为[0，1]，限制了它的应用范围。本文讨论F(x)的泛函形式未知，但经过logit变换后可用多项式逼近的情形，这就去掉了关于支撑集为[0，1]的限制。设F(x)有密度函数f(x)，且存在某个r≥1，(r未知)，使得logF(x)/(1-F(x))=a_0+a_1x+…+a_rx~r(?)J_r(x)其中x∈(a，b)，(a，b)(?)(a_0，b_0)。由最小二乘估计的思想，我们得到了逼近多项式的系数{a_0，a_1，…，a_r)的估计，进一步得到了分布函数F(x)的估计量，并且证明了系数估计{(?)_0，(?)_1，…，(?)_r}的一致收敛性和分布函数估计(?)(x)的一致收敛性。我们在主要讨论分布函数F(x)的同时，也得到了其密度函数f(x)和失效函数λ(x)的非参数估计及其相应的大样本结果。模拟结果显示估计是理想的。
本文最后作了总结与展望。
Varying coefficient model, also known as functional coefficient model, is auseful extension of the general linear model. Since varying coefficient model cancover a lot of other models, it has attracted the interest of many statistical researchers.Varying coefficient models not only retains partially the characteristics ofnonparametric regression, but is also simple in structure and easy to explain, whichhas led to its extensive application in statistical analysis in biology, medicine andmedicinal research, economics and finance.
This paper conducts research on varying coefficient model via wavelet method.As a technique, wavelet method was first used in the analysis of seismic data. With itstheoretical development, the method has been also found extensive application inimage processing, signal processing and data compressing. The early 1990s haswitnessed the application of wavelet method in statistical studies. Since waveletmethod provides a powerful new technique for statistical researchers and hasdemonstrated its various advantages, it has been highly valued, and recent years, it hasbeen rapid increase in the use of wavelet method in applied statistical studies.
This paper first studies the varying coefficient model by Hastie and Tibshirani(1993), and establishes unknown function coefficients and variance error estimator, and discusses asymptotic properties of estimators. The paper also discusses anothertype of nonparametric estimation. This research was financed by Fudan-SwitzerlandReinsurance Fund and the National Natural Science Foundation of China, and ourresearch findings have provided a practical method for research in mutual insurancefor the elderly. This paper tries to combine theory with practice, and realizes the crossconnection of different disciplines. The major contents of each chapter are as follows:
Chapter One summarizes the development and evolution of statistical models, withemphasis on variations of varying coefficient models and related estimation methods.Research achievements are outlined, and feasibility and extensiveness of the application of varying coefficient model are documented. Compared with othermethods, wavelet method has many advantages. For example, compared withorthogonal series estimation, wavelet method has a low demand on function, while thelarge sample resulting is more satisfactory. Wavelet method can accurately describethe local features of complex curves, which has caught the attention of manyengineers and researchers in mathematics and statistics. This chapter will give ageneral instruction to wavelet method so as to lay a basis for Chapters Two and Threein which wavelet method will be employed in analysis. At the end of this chapter, major research contents and results are elaborated, and a summary is given.
Chapter Two discusses the following varying coefficient model: y=x_1β_1(t)+…+x_pβ_p(t)+ewhere y is response variable, x=(x_1,…, x_p)~T and t are covariables, e is randomerror, whereas E(e)=0, E(e~2)=σ~2. {β-j(t), j=1,…, p} are unknownnonparametric functions, without losing generality, let the domain ofβ_j(t) be [0, 1].σ~2 is unknown parameter. This model can be seen as an extension of general linearmodel, which allows regression coefficient to be the function of a regression variablet.β_j(·) is able to describe in detail the relationship between x_j and t. In addition, sinceβ_j(·) is a general nonparametrie function, the error of regression model isnoticeably reduced, hence strongly avoid the phenomenon of "curse of dimension". Inthis paper, let x be random design, while t be fixed design. Apply nonparametricregression wavelet estimation to the afore-mentioned varying coefficient model, wecan establish wavelet estimation of function coefficients {β_j(t), j=1,…, p}, and canget strong uniform convergence and asymptotic normality.
Chapter Three discusses with error variance estimation of varying coefficientmodel mentioned previously, proposes wavelet estimation (?)_n~2 for error varianceσ~2, and proves its large sample nature. It also establishes wavelet estimation (?)_n~2 for var(e~2), and proves the weak consistence of (?)_n~2, from whichn~(1/2)((?)_n~2-σ~2)/(?)_n(?)N(0, 1) is deduced. This result can be used to construct largesample interval estimation or conduct large sample test forσ~2. For same model, compared with existing research results, this method has a weak demand for functionsto be estimated while achieving more satisfactory estimation property. In addition, theresearch results here can be used for statistical purposes, so it has practicalapplications.
Chapter Four discusses the estimation of survival functions in "mutual insurancefor the elderly project". The project stems from intensive computation of insurance.In mutual insurance for the elderly, we need to conduct estimation for survivalfunctions of self-sustaining life span, semi self-sustaining life span andnon-self-sustaining life span. The only know information is the sample of individualstate after the age of 60. What needs to be estimated is the survival functions of agiven life period prior to 60. What is special is that the sample-taking interval isdifferent from that for survival function, therefore traditional empirical distribution isnot totally applicable.
Let X_1, X_2,…, X_n be independent sample from F(x), when functional form ofF(x) is unknown but it can be approximated with polynomial on [0, 1]θ_1X±…±θ_rX~r, Zheng Zukang (1995) proposed the mixed moment estimation andestablished strong consistence of estimators. But Zheng Zukang's (1995) approachassumes that the support set for distribution function is on [0, 1], this has limited itsscope of application. This paper discusses unknown function form F(x) which canbe approximated after logit conversion, thereby eliminates the limit of support set[0, 1]. Let F(x) has density function f(x), and there exists a certain r≥1, (r isunknown) which satisfies log F(x)/1-F(x)=a_0+a_1x+…+a_rx~r(?)J_r(x)where x∈(a, b), (a, b)(?)(a_0, b_0). According to least square estimation theory, wecan get the estimation of approximating polynomial coefficients {a_0, a_1,…, a_r}, onwhich we get the estimators for distribution function F(x), and prove the uniformconvergence of coefficient estimation {(?)_0, (?)_1,…, (?)_r} and distribution function (?)(x).While discussing distribution function F(x), we also get the nonparametricestimation and corresponding large sample results for its density function f(x)andfailure functionλ(x). The simulation results indicate that estimation is satisfactory.
Last chapter outlines research achievements and assumes related studies in thefuture.
引文
[1] Hastie, T. and Tibshirani, R., Varying-coefficient model[J], J. R. Statist. Soc. Ser.B. 1993, 55: 757-796.
[2] Bellman, R. E., Adaptive Control Processes, Princeton, Princeton University Press,1961.
[3] Silverman, B. J., Density estimation for statistics and Data Analysis, London,Chapman and Hall, 1986.
[4] Hardle, W. P., Applied Nonparametric Regression, Boston, Cambridge University Press, 1990.
[5] Hastie, T.J. and Tisbshirani, R., Generalized additive models, London: Chapman and Hall, 1990.
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