计量经济模型中非参数M估计的渐近理论
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摘要
在过去的几十年中,越来越多的研究者用非参数方法来对统计模型中的回归函数进行估计。很多估计方法以及估计量都已经被提出并得到了发展,如核估计,样条估计,局部回归估计以及正交序列估计方法等。非参数估计的理论以及实际应用都已经得到了系统的研究。至今,非参数估计仍然是统计中的一个热门与活跃的领域。在本文中我们将研究一类稳健的非参数估计:M估计。与其他类型的非参数估计量(如非参数最小二乘估计量)相比较,M估计量有以下的优点:它们对于异常点是稳健的,并且即使当观测值被污染或者残差是重尾分布时,它们仍然有很好的性质。
M估计是由Huber在1964年提出并用于位置参数的估计。M估计是一类稳健的估计。并且如Huber在1973所指出,当涉及到渐近理论时,M估计比其他的稳健估计(如L估计以及R估计)更容易处理。自从被提出以来,M估计的方法不论是在参数情形还是在非参数情形中都得到了深入的研究。不仅如此,一些学者还提出了改良化的M估计量。这些改良化的M估计量不仅继承了M估计量本身的优点,而且还具备了其他一些估计量的良好性质。例如,局部M估计量就是将局部线性光滑化方法与M估计的方法相结合后所产生的。因此,局部M估计量继承了局部多项式估计的优点并且克服了其非稳健性的缺点。我们将在第二章中研究相依空间过程的非参数回归函数及其导数的局部M估计。
过去对非参数M估计的研究大多针对于时间序列。对空间数据(或随机场)的稳健估计的研究相对比较少。然而在近几年中,越来越多的人开始关注空间数据的建模。这是因为空间数据在很多领域中都有广泛的应用,如经济学,流行病学,环境科学,图像分析以及海洋学等。因此在本文中,我们首先探讨一些相依空间数据的非参数M回归估计的渐近理论。在§2.1中,我们得到了相伴随机场非参数回归函数及其导数的局部M估计量的弱相合性以及渐近正态分布。在本节中,由于我们需要运用Bulinski引理来计算一些相伴随机场变量的非线性函数的协方差,所以我们对损失函数的导数ψ加了相对较强的限制条件。在§2.2中,我们建立了一个空间固定设计模型中回归函数及其导数的局部M估计量的弱相合性,强相合性以及渐近分布。该节中的空间过程满足一定的混合条件。由于§2.1以及之前一些文献中的损失函数ρ及其导数ψ都需要满足一些较为苛刻的条件,这使得一些重要的特殊例子都被排除在外。而我们在§2.2中所使用的方法则使得ρ与ψ的条件大为减弱。我们所考虑的ρ函数涵盖了此前的大部分作者所考虑的ρ。在§2.3中,我们建立了混合空间过程的非参数回归函数及其导数的局部M估计量的强Bahadur表示式。由此表示式,我们可以得到该局部M估计量的强相合性以及渐近正态分布。在§2.4中,我们用Monte-Carlo试验来说明第二章中所研究的局部M估计量的表现。由于我们一般不能通过定义局部M估计量的估计方程直接得到该估计量的明确表达式,所以我们采取了一个迭代的过程来推导该估计量。模拟结果显示,我们的估计方法在处理被污染或者重尾残差时的效果比NW(Nadaraya-Watson)估计量要好得多。
随着科学技术的发展,数据收集与测量的手段和方法也在不断进步,因此在实际应用中我们经常需要处理泛函型数据(如随机曲线)。泛函数据分析在很多领域,如犯罪学,经济学以及神经生理学,都有重要的应用。因此在最近几年中,越来越多的研究者开始关注泛函数据的建模与分析。在第三章中,我们将考虑混合泛函型数据的非参数回归函数的M估计。此章中我们所考虑的回归变量取值于某一抽象的半度量空间(例如R~d空间,Banach空间以及Hilbert空间),而响应变量则为实值随机变量。我们提出用非参数M估计的方法来对定义于抽象泛函空间的回归函数进行估计。我们建立了该M估计量的渐近相合性以及渐近分布。我们所要求的关于损失函数ρ及其导数ψ的条件在此类问题的研究是比较弱的,这使得我们的结果包括了一些重要的估计量,如最小绝对距离估计量,混合最小二乘与最小绝对距离估计量。另外我们还给出了两个满足第三章中混合条件的泛函序列的例子。最后,我们用Monte-Carlo模拟来说明我们的方法能很好地处理重尾残差。
在第四章中,我们考虑一个固定设计回归模型。在这个模型中,残差为一个长程相依的线性过程。我们用非参数M估计量来对模型中的回归函数进行估计,并得到了该M估计量的渐近一阶以及渐近二阶展开。我们将所得到的结果与NW估计量进行了比较,通过比较我们发现:非参数M估计量与NW估计量是渐近一阶等价的,这表明M估计量与NW估计量有相同的渐近分布。另外我们还证明了非参数M估计量与NW估计量之差在适当的标准化后存在着极限分布,这一极限分布与长程相依的参数α有关。我们通过一个模拟试验来比较非参数M估计量与NW估计量的有限样本性质。我们通过两个残差为长程相依线性过程的固定设计模型来比较这两个估计量的均方误差。此外,我们还画出了这两个估计量的轨迹。从模拟结果可以发现,与NW估计量相比较,非参数M估计量对污染数据是稳健的。
在本文第二至第四章中所涉及的随机样本都被假设为是平稳的。然而由于在计量经济以及金融中存在着很多的非平稳数据,例如价格以及汇率,所以在第五章中我们研究一类非平稳变量的非参数回归估计。单位根过程是一类在计量经济中有重要应用的非平稳过程,所以在此章中我们考虑共变量是单位根过程的一个非线性共积分模型。我们建立了该非线性共积分模型的回归函数的M估计量的弱相合性以及渐近分布。该渐近分布是混合正态的,并且不同于平稳时间序列的相关结果。从我们所得到的结果可以发现,第五章中所考虑的非平稳时间序列的非参数M估计量的收敛速度比平稳时间序列的收敛速度要慢,而这也正是我们所预期的,因为非平稳随机样本落在某一固定点的邻域中的观测值比平稳时间序列要少。在§5.3的模拟中,我们依然用迭代方法来推导非平稳数据的非参数回归函数的M估计量。我们分别给出了三个例子并进行了Monte-Carlo试验。通过比较非参数M估计量与对应的NW估计量的表现,我们可以看出:当残差被污染或者是重尾分布时,M估计量的稳健性比NW估计量要好得多。
In the last several decades,there has been much interest in nonparametric estimation of statistical regression functions.Various estimation methods and estimators have been proposed and well developed,such as kernel,spline,local regression and orthogonal series methods.Both the theory and practical implementation of nonparametric estimates have been systematically studied.And now it is still a hot topic and an active field in statistics.In this thesis,we study a type of robust nonparametric estimation:nonparametric M estimation.Compared to other types of nonparametric estimators(such as the nonparametric least-squares estimators),M-estimators have the following advantages: they are robust to outliers and they perform well even when the observations are contaminated or the errors are heavy-tailed.
M estimation was first introduced by Huber in 1964 to estimate a location parameter. It is one dass of the robust estimates.As pointed out by Huber in 1973,compared to other types of robust estimates,such as L-estimates and R-estimates,M-estimates are easiest to cope with as far as asymptotic theory is concerned.Since its introduction, M-estimation has been studied in depth by many authors,not only in parametric setting but also in nonparametric setting.Furthermore,some authors have proposed some modified M-estimators,which inherit many nice statistical properties from the M-estimators and other types of estimators.For example,the local M-estimators are obtained by a combination of the local linear smoothing technique and the M-estimation technique.So they inherit the advantages of local polynomial smoothers and overcome their shortcoming of lack of robustness.We will study the local M-estimation of nonparametric regression functions and their derivatives for dependent spatial data in Chapter 2.
Previous work on nonparametric M estimation is mostly concentrated on time series. Studies on robust estimation for spatial processes(or random fields) are comparatively few.However,there is increasing interest in spatial data modelling,as it has wide applications in many fields,such as econometrics,epidemiology,environmental science,image analysis,oceanography et al.So in this thesis,we first explore the asymptotic theory of nonparametric M estimation for certain dependent spatial processes.The spatial data in our thesis satisfy certain dependent structure,such as association and mixing.In§2.1,we obtain the weak consistency and asymptotic normality of the spatial local M-estimators of the nonparametric regression function and its derivative for associated processes.We impose relatively strong restrictions on theψ-function(the derivative of the loss function) in this section,as we need to apply Bulinski's Lemma to bound the covariances of certain nonlinear functions of blocks of associated random variables.In§2.2,we derive the weak and strong consistency as well as asymptotic distribution of the local M-estimators for a spatial fixed-design model.The spatial data in this section satisfy a mixing condition.As the conditions on the loss functionρand its derivativeψin§2.1 and many other papers are restrictive,which do not cover some important special cases,we apply a method that can greatly weaken these conditions.The functionρconsidered in§2.2 covers most of theρfunctions considered by earlier writers. In§2.3,we establish the strong Bahadur representation of the local M-estimators of the nonparametric regression function and its derivative for mixing spatial processes. From this representation,we can obtain the strong consistency of the local M-estimators as well as their asymptotic distribution.In§2.4,we implement Monte-Carlo experiments to show the behavior of the local M-estimators considered in Chapter 2.As the local M-regression estimators are defined implicitly through an estimating equation,we adopt an iterative scheme to derive the estimators.The simulation results show that our methods behave much better than the NW(Nadaraya-Watson) estimators when dealing with contaminated or heavy-tailed errors.
Due to developments in means of data collection,we often need to deal with functional data(such as random curves) in practice.Function data analysis has a lot of applications in many areas,such as criminology,economics and neurophysiology et al.So there has arisen much interest in functional data modelling and analysis in recent years. In Chapter 3,we focus on M estimation for mixing functional data.The regressors in this chapter take values in some abstract semi-metric space(such as R~d space,Banach space and Hilbert space) and the response variables are real-valued random variables. We propose a nonparametric M-estimator to estimate the regression function that is defined in the abstract space and establish asymptotic consistency and distribution of the estimator.The conditions on the loss functionρand its derivativeψare relatively mild in this kind of problems and cover many important estimators,such as least-absolute distance estimators and mixed least-squares estimators and least-absolute distance estimators. We also give two examples of multivariate time series that satisfy the mixing conditions in this chapter.Furthermore,we implement a Monte-Carlo experiment to show that out method can cope well with heavy-tailed random errors.
In Chapter 4,we consider a fixed-design regression model,where the error is a long-range dependent linear process.We derive the first order and second order expansion of the proposed M estimator and compare the M estimator with corresponding NW estimator.We find that the nonparametric M-estimator is first-order equivalent to the NW estimator,which implies that the nonparametric M-estimator has the same asymptotic distribution as that of the NW estimator.Furthermore,we show that the difference of the nonparametric M-estimator and the NW estimator has a limiting distribution after suitable standardization.The nature of the limiting distribution depends on the range of long-memory parameterα.We also illustrate the finite sample behavior of the nonparametric M estimator and the NW estimator by performing a simulation study.We compare the mean squared error of the two estimators through two examples of fixed-design models with long-range dependent linear errors.Furthermore,we depict the path of the two estimators.From the simulation,we can find that,compared to the NW estimator,the nonparametric M estimator is robust to contaminated data.
The observations involved in the foregoing three chapters are all assumed to be stationary.As there are numerous nonstationary data in econometrics and finance,such as price and exchange rate,we study nonparametric regression estimation for a class of nonstationary time series in Chapter 5.We consider a nonlinear cointegration model with unit root type covariates,which have many applications in econometrics.We establish the weak consistency and asymptotic distribution for the proposed nonparametric M estimator.The asymptotic distribution turns out to be mixed normal and be different from that for stationary time series.From the results we establish,we know that the convergence rate of the M estimator for the nonstationary time series that are considered in this chapter is slower than that for stationary time series.This is not hard to understand,as there are less observations falling in the neighborhood of any fixed point for nonstationary time series than for stationary time series.As it is not so easy to obtain the M estimator directly from the estimating equation,we still adopt an iterative procedure to derive the estimator.We then use three examples and implement Monte- Carle experiments to show that our method work well in practice.We achieve this by comparing the performance of the proposed M estimator and the corresponding NW estimator.The results of the experiments show that when the errors are contaminated or heavy-tailed,the M estimator outperforms the NW estimator.
M估计是由Huber在1964年提出并用于位置参数的估计。M估计是一类稳健的估计。并且如Huber在1973所指出,当涉及到渐近理论时,M估计比其他的稳健估计(如L估计以及R估计)更容易处理。自从被提出以来,M估计的方法不论是在参数情形还是在非参数情形中都得到了深入的研究。不仅如此,一些学者还提出了改良化的M估计量。这些改良化的M估计量不仅继承了M估计量本身的优点,而且还具备了其他一些估计量的良好性质。例如,局部M估计量就是将局部线性光滑化方法与M估计的方法相结合后所产生的。因此,局部M估计量继承了局部多项式估计的优点并且克服了其非稳健性的缺点。我们将在第二章中研究相依空间过程的非参数回归函数及其导数的局部M估计。
过去对非参数M估计的研究大多针对于时间序列。对空间数据(或随机场)的稳健估计的研究相对比较少。然而在近几年中,越来越多的人开始关注空间数据的建模。这是因为空间数据在很多领域中都有广泛的应用,如经济学,流行病学,环境科学,图像分析以及海洋学等。因此在本文中,我们首先探讨一些相依空间数据的非参数M回归估计的渐近理论。在§2.1中,我们得到了相伴随机场非参数回归函数及其导数的局部M估计量的弱相合性以及渐近正态分布。在本节中,由于我们需要运用Bulinski引理来计算一些相伴随机场变量的非线性函数的协方差,所以我们对损失函数的导数ψ加了相对较强的限制条件。在§2.2中,我们建立了一个空间固定设计模型中回归函数及其导数的局部M估计量的弱相合性,强相合性以及渐近分布。该节中的空间过程满足一定的混合条件。由于§2.1以及之前一些文献中的损失函数ρ及其导数ψ都需要满足一些较为苛刻的条件,这使得一些重要的特殊例子都被排除在外。而我们在§2.2中所使用的方法则使得ρ与ψ的条件大为减弱。我们所考虑的ρ函数涵盖了此前的大部分作者所考虑的ρ。在§2.3中,我们建立了混合空间过程的非参数回归函数及其导数的局部M估计量的强Bahadur表示式。由此表示式,我们可以得到该局部M估计量的强相合性以及渐近正态分布。在§2.4中,我们用Monte-Carlo试验来说明第二章中所研究的局部M估计量的表现。由于我们一般不能通过定义局部M估计量的估计方程直接得到该估计量的明确表达式,所以我们采取了一个迭代的过程来推导该估计量。模拟结果显示,我们的估计方法在处理被污染或者重尾残差时的效果比NW(Nadaraya-Watson)估计量要好得多。
随着科学技术的发展,数据收集与测量的手段和方法也在不断进步,因此在实际应用中我们经常需要处理泛函型数据(如随机曲线)。泛函数据分析在很多领域,如犯罪学,经济学以及神经生理学,都有重要的应用。因此在最近几年中,越来越多的研究者开始关注泛函数据的建模与分析。在第三章中,我们将考虑混合泛函型数据的非参数回归函数的M估计。此章中我们所考虑的回归变量取值于某一抽象的半度量空间(例如R~d空间,Banach空间以及Hilbert空间),而响应变量则为实值随机变量。我们提出用非参数M估计的方法来对定义于抽象泛函空间的回归函数进行估计。我们建立了该M估计量的渐近相合性以及渐近分布。我们所要求的关于损失函数ρ及其导数ψ的条件在此类问题的研究是比较弱的,这使得我们的结果包括了一些重要的估计量,如最小绝对距离估计量,混合最小二乘与最小绝对距离估计量。另外我们还给出了两个满足第三章中混合条件的泛函序列的例子。最后,我们用Monte-Carlo模拟来说明我们的方法能很好地处理重尾残差。
在第四章中,我们考虑一个固定设计回归模型。在这个模型中,残差为一个长程相依的线性过程。我们用非参数M估计量来对模型中的回归函数进行估计,并得到了该M估计量的渐近一阶以及渐近二阶展开。我们将所得到的结果与NW估计量进行了比较,通过比较我们发现:非参数M估计量与NW估计量是渐近一阶等价的,这表明M估计量与NW估计量有相同的渐近分布。另外我们还证明了非参数M估计量与NW估计量之差在适当的标准化后存在着极限分布,这一极限分布与长程相依的参数α有关。我们通过一个模拟试验来比较非参数M估计量与NW估计量的有限样本性质。我们通过两个残差为长程相依线性过程的固定设计模型来比较这两个估计量的均方误差。此外,我们还画出了这两个估计量的轨迹。从模拟结果可以发现,与NW估计量相比较,非参数M估计量对污染数据是稳健的。
在本文第二至第四章中所涉及的随机样本都被假设为是平稳的。然而由于在计量经济以及金融中存在着很多的非平稳数据,例如价格以及汇率,所以在第五章中我们研究一类非平稳变量的非参数回归估计。单位根过程是一类在计量经济中有重要应用的非平稳过程,所以在此章中我们考虑共变量是单位根过程的一个非线性共积分模型。我们建立了该非线性共积分模型的回归函数的M估计量的弱相合性以及渐近分布。该渐近分布是混合正态的,并且不同于平稳时间序列的相关结果。从我们所得到的结果可以发现,第五章中所考虑的非平稳时间序列的非参数M估计量的收敛速度比平稳时间序列的收敛速度要慢,而这也正是我们所预期的,因为非平稳随机样本落在某一固定点的邻域中的观测值比平稳时间序列要少。在§5.3的模拟中,我们依然用迭代方法来推导非平稳数据的非参数回归函数的M估计量。我们分别给出了三个例子并进行了Monte-Carlo试验。通过比较非参数M估计量与对应的NW估计量的表现,我们可以看出:当残差被污染或者是重尾分布时,M估计量的稳健性比NW估计量要好得多。
In the last several decades,there has been much interest in nonparametric estimation of statistical regression functions.Various estimation methods and estimators have been proposed and well developed,such as kernel,spline,local regression and orthogonal series methods.Both the theory and practical implementation of nonparametric estimates have been systematically studied.And now it is still a hot topic and an active field in statistics.In this thesis,we study a type of robust nonparametric estimation:nonparametric M estimation.Compared to other types of nonparametric estimators(such as the nonparametric least-squares estimators),M-estimators have the following advantages: they are robust to outliers and they perform well even when the observations are contaminated or the errors are heavy-tailed.
M estimation was first introduced by Huber in 1964 to estimate a location parameter. It is one dass of the robust estimates.As pointed out by Huber in 1973,compared to other types of robust estimates,such as L-estimates and R-estimates,M-estimates are easiest to cope with as far as asymptotic theory is concerned.Since its introduction, M-estimation has been studied in depth by many authors,not only in parametric setting but also in nonparametric setting.Furthermore,some authors have proposed some modified M-estimators,which inherit many nice statistical properties from the M-estimators and other types of estimators.For example,the local M-estimators are obtained by a combination of the local linear smoothing technique and the M-estimation technique.So they inherit the advantages of local polynomial smoothers and overcome their shortcoming of lack of robustness.We will study the local M-estimation of nonparametric regression functions and their derivatives for dependent spatial data in Chapter 2.
Previous work on nonparametric M estimation is mostly concentrated on time series. Studies on robust estimation for spatial processes(or random fields) are comparatively few.However,there is increasing interest in spatial data modelling,as it has wide applications in many fields,such as econometrics,epidemiology,environmental science,image analysis,oceanography et al.So in this thesis,we first explore the asymptotic theory of nonparametric M estimation for certain dependent spatial processes.The spatial data in our thesis satisfy certain dependent structure,such as association and mixing.In§2.1,we obtain the weak consistency and asymptotic normality of the spatial local M-estimators of the nonparametric regression function and its derivative for associated processes.We impose relatively strong restrictions on theψ-function(the derivative of the loss function) in this section,as we need to apply Bulinski's Lemma to bound the covariances of certain nonlinear functions of blocks of associated random variables.In§2.2,we derive the weak and strong consistency as well as asymptotic distribution of the local M-estimators for a spatial fixed-design model.The spatial data in this section satisfy a mixing condition.As the conditions on the loss functionρand its derivativeψin§2.1 and many other papers are restrictive,which do not cover some important special cases,we apply a method that can greatly weaken these conditions.The functionρconsidered in§2.2 covers most of theρfunctions considered by earlier writers. In§2.3,we establish the strong Bahadur representation of the local M-estimators of the nonparametric regression function and its derivative for mixing spatial processes. From this representation,we can obtain the strong consistency of the local M-estimators as well as their asymptotic distribution.In§2.4,we implement Monte-Carlo experiments to show the behavior of the local M-estimators considered in Chapter 2.As the local M-regression estimators are defined implicitly through an estimating equation,we adopt an iterative scheme to derive the estimators.The simulation results show that our methods behave much better than the NW(Nadaraya-Watson) estimators when dealing with contaminated or heavy-tailed errors.
Due to developments in means of data collection,we often need to deal with functional data(such as random curves) in practice.Function data analysis has a lot of applications in many areas,such as criminology,economics and neurophysiology et al.So there has arisen much interest in functional data modelling and analysis in recent years. In Chapter 3,we focus on M estimation for mixing functional data.The regressors in this chapter take values in some abstract semi-metric space(such as R~d space,Banach space and Hilbert space) and the response variables are real-valued random variables. We propose a nonparametric M-estimator to estimate the regression function that is defined in the abstract space and establish asymptotic consistency and distribution of the estimator.The conditions on the loss functionρand its derivativeψare relatively mild in this kind of problems and cover many important estimators,such as least-absolute distance estimators and mixed least-squares estimators and least-absolute distance estimators. We also give two examples of multivariate time series that satisfy the mixing conditions in this chapter.Furthermore,we implement a Monte-Carlo experiment to show that out method can cope well with heavy-tailed random errors.
In Chapter 4,we consider a fixed-design regression model,where the error is a long-range dependent linear process.We derive the first order and second order expansion of the proposed M estimator and compare the M estimator with corresponding NW estimator.We find that the nonparametric M-estimator is first-order equivalent to the NW estimator,which implies that the nonparametric M-estimator has the same asymptotic distribution as that of the NW estimator.Furthermore,we show that the difference of the nonparametric M-estimator and the NW estimator has a limiting distribution after suitable standardization.The nature of the limiting distribution depends on the range of long-memory parameterα.We also illustrate the finite sample behavior of the nonparametric M estimator and the NW estimator by performing a simulation study.We compare the mean squared error of the two estimators through two examples of fixed-design models with long-range dependent linear errors.Furthermore,we depict the path of the two estimators.From the simulation,we can find that,compared to the NW estimator,the nonparametric M estimator is robust to contaminated data.
The observations involved in the foregoing three chapters are all assumed to be stationary.As there are numerous nonstationary data in econometrics and finance,such as price and exchange rate,we study nonparametric regression estimation for a class of nonstationary time series in Chapter 5.We consider a nonlinear cointegration model with unit root type covariates,which have many applications in econometrics.We establish the weak consistency and asymptotic distribution for the proposed nonparametric M estimator.The asymptotic distribution turns out to be mixed normal and be different from that for stationary time series.From the results we establish,we know that the convergence rate of the M estimator for the nonstationary time series that are considered in this chapter is slower than that for stationary time series.This is not hard to understand,as there are less observations falling in the neighborhood of any fixed point for nonstationary time series than for stationary time series.As it is not so easy to obtain the M estimator directly from the estimating equation,we still adopt an iterative procedure to derive the estimator.We then use three examples and implement Monte- Carle experiments to show that our method work well in practice.We achieve this by comparing the performance of the proposed M estimator and the corresponding NW estimator.The results of the experiments show that when the errors are contaminated or heavy-tailed,the M estimator outperforms the NW estimator.
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