线性模型参数的约束有偏估计和预检验估计研究
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摘要
线性模型在现代统计方法中占重要地位,是现代统计学中应用最为广泛的一类模型之一。本文主要是研究线性模型参数在等式约束和随机线性约束下的约束有偏估计以及预检验估计等相关的一些问题。
对于等式约束下的线性模型,通过分别综合岭估计,Liu估计和约束最小二乘估计,本文提出了一类新的严格满足等式约束条件的约束岭估计和约束Liu估计,并证明了在均方误差矩阵意义下它们将分别优于传统的岭估计,Liu估计和约束最小二乘估计。此外,本文也把文献中研究较多的r-k估计和r-d估计推广到等式约束情形,即分别研究了约束r-k估计和约束r-d估计,并同样证明了在均方误差矩阵意义下,约束r-k估计和r-d估计都将分别优于r-k估计和r-d估计。而对于最近几年提出的双参数Liu-Type估计,本文对调节参数的选取及其他一些统计拟合性质做了进一步的研究。具体来讲,本文从两个角度来讨论了最优调节参数的选取问题,即最大化复相关系数和最小化广义交叉验证标准。此外,证明了在Liu-Type估计中,在通过适当选取岭参数改善设计阵的病态性带来的影响的同时,恰当地选取调节参数能够有效地改进回归的效果。而随着岭参数的增加,Liu-Type估计将比传统的岭估计更具稳健性。此外,通过实例分析,验证了得到的理论结果。对于随机约束下的线性模型,本文把传统的混合估计推广到了奇异线性模型情形,即提出了奇异混合估计,并对其均方误差矩阵优良性和两步估计分别进行了讨论。通过把奇异混合估计应用于随机约束下的Panel数据模型参数估计之中,我们导出了Panel数据模型在随机约束下固定效应的四种可行估计,并对它们之间的关系以及与相应的无约束估计之间的优良性进行了详细比较。此外,本文把奇异混合估计应用于奇异线性模型的预测理论之中,研究了奇异线性模型的最优线性非齐次无偏预测,最优线性齐次无偏预测以及随机约束下的最优线性无偏预测,并证明了它们都满足一个一般的预测通式。通过综合混合估计的思想和Liu估计,本文提出了一类新的随机约束Liu估计,并证明了在一定的条件下,它将分别优于传统的Liu估计,混合估计,并通过模拟研究和实例分析,验证了理论上得到的结果。针对文献中常见的两类错误指定模型的情况,本文进一步研究了随机约束Liu估计在这两类情形下的优良性,并对相应的经典预测的表现也进行了考察。
对于等式约束预检验估计的研究,本文首先对文献中常见的一些检验方法进行了讨论,包括传统的F检验以及一些在计量经济类模型中应用广泛的大样本检验方法,比如Wald检验,LR检验以及LM检验。通过综合Liu估计和预检验估计的思想,本文提出了基于上述三个大样本检验的等式约束预检验Liu估计。通过对偏差进行分析,我们发现基于Wald检验的等式约束预检验Liu估计有最小的平方偏差,其次为基于LR检验和LM检验的相应估计。通过对均方误差的分析,我们发现在原假设附近,基于LM检验的预检验Liu估计有最小的均方误差,其次为基于LR检验和Wald检验的估计,而当偏离原假设时,情况刚好相反;另一方面,在Liu参数取值较小的时候,基于Wald检验的估计最优,其次为基于LR检验和LM检验的估计,而在Liu参数取值较大时,情况也恰好相反。此外,对这三个基于大样本检验的等式约束预检验Liu估计,我们还对其相对效率以及基于极大极小相对效率准则的最优显著性水平的选取问题分别进行了讨论。考虑到实际数据可能存在的厚尾情况,本文研究了文献中讨论颇多的多元t分布模型的基于大样本检验的等式约束预检验Liu估计,并同样在平方偏差和均方误差两个准则下对其优良性进行了详细比较。
最后,本文研究了随机约束下的参数预检验估计。通过把随机约束转化成等式约束的框架下来处理,研究了基于F检验的随机约束预检验岭估计。在均方误差准则下对随机约束预检验岭估计,随机约束预检验估计以及相应的岭估计的优良性进行了系统分析,并对随机约束预检验估计的相对效率和相应最优显著性水平的选择进行了研究。
Linear models play a central part in modern statistical methods and they have become one of the most widely used classes of models in modern statistics. In this dissertation, we mainly focused on the restricted biased estimation and preliminary test estimation of parameter in linear models with equality restrictions and stochastic linear restrictions.
For linear model with exact linear equality restrictions, by respectively combining the ridge estimator and Liu estimator with the restricted least squares estimator (RLSE), a new class of restricted ridge estimator and restricted Liu estimator which always satisfy the given linear restrictions are proposed. The new restricted ridge estimator is proved to be superior to the traditional ridge estimator and the RLSE, while the new restricted Liu estimator outperforms Liu estimator and the RLSE in the sense of mean squared error matrix (MSEM). In addition, this paper generalize the r-k estimator and r-d estimator proposed recently in literatures to the cases with restrictions, namely we have studied the restricted r-k estimator and restricted r-d estimator in this paper, and we similarly prove that the restricted r-k estimator and the restricted r-d estimator outperforms the r-k estimator and r-d estimator in the MSEM sense, respectively. For the Liu-Type estimator considered by many researchers in literatures recently, we have made a further discussion about the choice of the tuning parameter and some other fitting characteristics. In particular, we derived two methods to determine the optimum tuning parameter, which are to maximize the coefficient of multiple determination or to minimize the generalized cross validation (GCV) of the prediction quality. It’s proved that for the Liu-Type estimator, the ridge parameter could serve for regularization of an ill-conditioned design matrix, while the tuning parameter could be used for tuning the fit quality effectively, and as the ridge parameter increases, the Liu-Type estimator produces more robust regression models than the ridge estimator. Numerical examples are given to illustrate the theoretical results.
For linear model with stochastic linear restrictions, in this thesis we generalize the traditional ordinary mixed estimator (OME) to singular linear model and proposed singular mixed estimator (SME). Performances of the SME in the MSEM sense and its two-stage estimator are also discussed. Applying the SME in the parameter estimation of Panel data model with stochastic linear restrictions, we derived four feasible estimators for Panel data model. The relationship among them is discussed and it’s shown that they are superior to the corresponding unrestricted estimators. Furthermore, we have studied the prediction problem in singular linear model based on the SME. The optimal heterogeneous predictor, optimal unbiased homogeneous predictor and the optimal predictor for singular linear model with stochastic linear restrictions are derived, and we find that all the three predictors satisfy a general formula for prediction. By combining the idea of mixed estimation and Liu estimator, we proposed a new stochastic restricted Liu estimator and prove that it outperforms the traditional Liu estimator, OME and some other stochastic restricted estimators under certain conditions. Simulation study and numerical example have supported the theoretical results derived. For two common types of misspecification of regression models, we have further studied the behaviors of the stochastic restricted Liu estimator in such two cases, and performances of the corresponding predictors are also examined.
For the preliminary test estimator when exact linear restrictions are used, we have firstly discussed some common test methods in literatures, including the familiar F test and some large sample tests widely used in Econometric models such as the Wald test, Likelihood Ratio (LR) test and Lagrangian Multiplier (LM) test. By combing the idea of preliminary test and Liu estimator, we have proposed three preliminary test Liu estimators (PTLE) based on the Wald, LR and LM tests. Through the bias analysis, we find that the Wald test based PTLE has the smallest quadratic bias,followed by the PTLE based on the LR and LM tests. On the other hand, we find that when near the null hypothesis, the LM test based PTLE has the smallest risk, followed by the PTLE based on the LR and Wald tests, while when the parameter departs from the restrictions, the situation is reversed. In the meantime, when the Liu parameter is small, then the PTLE based on W test has the smallest MSE followed by the LR and LM tests, while when the Liu parameter is large and near one, the situation is just also reversed. Furthermore, the relative efficiency and the choice of optimal significance levels are also discussed. Considering the fat-tail phenomenon that may exist in practical data, we have also studied the PTLE for the multi-t distribution model based on the three large sample tests above. Performances of the estimators according to the quadratic bias and mean square error are similarly compared in detail.
Finally, we have also considered the preliminary test estimator when stochastic restrictions are used in regression. By imbedding the stochastic linear restriction model in an exact linear restriction framework and make use of the results concerning the quality restricted estimator, we propose the stochastic hypothesis preliminary test ridge estimator (SPTRE) based on F test. Performances of the SPTRE, stochastic hypothesis preliminary test estimator and the ridge estimator in the sense of MSE are systematically analyzed. Relative efficiency of the SPRE and the corresponding optimal choice of the significance level are also discussed in this paper.
对于等式约束下的线性模型,通过分别综合岭估计,Liu估计和约束最小二乘估计,本文提出了一类新的严格满足等式约束条件的约束岭估计和约束Liu估计,并证明了在均方误差矩阵意义下它们将分别优于传统的岭估计,Liu估计和约束最小二乘估计。此外,本文也把文献中研究较多的r-k估计和r-d估计推广到等式约束情形,即分别研究了约束r-k估计和约束r-d估计,并同样证明了在均方误差矩阵意义下,约束r-k估计和r-d估计都将分别优于r-k估计和r-d估计。而对于最近几年提出的双参数Liu-Type估计,本文对调节参数的选取及其他一些统计拟合性质做了进一步的研究。具体来讲,本文从两个角度来讨论了最优调节参数的选取问题,即最大化复相关系数和最小化广义交叉验证标准。此外,证明了在Liu-Type估计中,在通过适当选取岭参数改善设计阵的病态性带来的影响的同时,恰当地选取调节参数能够有效地改进回归的效果。而随着岭参数的增加,Liu-Type估计将比传统的岭估计更具稳健性。此外,通过实例分析,验证了得到的理论结果。对于随机约束下的线性模型,本文把传统的混合估计推广到了奇异线性模型情形,即提出了奇异混合估计,并对其均方误差矩阵优良性和两步估计分别进行了讨论。通过把奇异混合估计应用于随机约束下的Panel数据模型参数估计之中,我们导出了Panel数据模型在随机约束下固定效应的四种可行估计,并对它们之间的关系以及与相应的无约束估计之间的优良性进行了详细比较。此外,本文把奇异混合估计应用于奇异线性模型的预测理论之中,研究了奇异线性模型的最优线性非齐次无偏预测,最优线性齐次无偏预测以及随机约束下的最优线性无偏预测,并证明了它们都满足一个一般的预测通式。通过综合混合估计的思想和Liu估计,本文提出了一类新的随机约束Liu估计,并证明了在一定的条件下,它将分别优于传统的Liu估计,混合估计,并通过模拟研究和实例分析,验证了理论上得到的结果。针对文献中常见的两类错误指定模型的情况,本文进一步研究了随机约束Liu估计在这两类情形下的优良性,并对相应的经典预测的表现也进行了考察。
对于等式约束预检验估计的研究,本文首先对文献中常见的一些检验方法进行了讨论,包括传统的F检验以及一些在计量经济类模型中应用广泛的大样本检验方法,比如Wald检验,LR检验以及LM检验。通过综合Liu估计和预检验估计的思想,本文提出了基于上述三个大样本检验的等式约束预检验Liu估计。通过对偏差进行分析,我们发现基于Wald检验的等式约束预检验Liu估计有最小的平方偏差,其次为基于LR检验和LM检验的相应估计。通过对均方误差的分析,我们发现在原假设附近,基于LM检验的预检验Liu估计有最小的均方误差,其次为基于LR检验和Wald检验的估计,而当偏离原假设时,情况刚好相反;另一方面,在Liu参数取值较小的时候,基于Wald检验的估计最优,其次为基于LR检验和LM检验的估计,而在Liu参数取值较大时,情况也恰好相反。此外,对这三个基于大样本检验的等式约束预检验Liu估计,我们还对其相对效率以及基于极大极小相对效率准则的最优显著性水平的选取问题分别进行了讨论。考虑到实际数据可能存在的厚尾情况,本文研究了文献中讨论颇多的多元t分布模型的基于大样本检验的等式约束预检验Liu估计,并同样在平方偏差和均方误差两个准则下对其优良性进行了详细比较。
最后,本文研究了随机约束下的参数预检验估计。通过把随机约束转化成等式约束的框架下来处理,研究了基于F检验的随机约束预检验岭估计。在均方误差准则下对随机约束预检验岭估计,随机约束预检验估计以及相应的岭估计的优良性进行了系统分析,并对随机约束预检验估计的相对效率和相应最优显著性水平的选择进行了研究。
Linear models play a central part in modern statistical methods and they have become one of the most widely used classes of models in modern statistics. In this dissertation, we mainly focused on the restricted biased estimation and preliminary test estimation of parameter in linear models with equality restrictions and stochastic linear restrictions.
For linear model with exact linear equality restrictions, by respectively combining the ridge estimator and Liu estimator with the restricted least squares estimator (RLSE), a new class of restricted ridge estimator and restricted Liu estimator which always satisfy the given linear restrictions are proposed. The new restricted ridge estimator is proved to be superior to the traditional ridge estimator and the RLSE, while the new restricted Liu estimator outperforms Liu estimator and the RLSE in the sense of mean squared error matrix (MSEM). In addition, this paper generalize the r-k estimator and r-d estimator proposed recently in literatures to the cases with restrictions, namely we have studied the restricted r-k estimator and restricted r-d estimator in this paper, and we similarly prove that the restricted r-k estimator and the restricted r-d estimator outperforms the r-k estimator and r-d estimator in the MSEM sense, respectively. For the Liu-Type estimator considered by many researchers in literatures recently, we have made a further discussion about the choice of the tuning parameter and some other fitting characteristics. In particular, we derived two methods to determine the optimum tuning parameter, which are to maximize the coefficient of multiple determination or to minimize the generalized cross validation (GCV) of the prediction quality. It’s proved that for the Liu-Type estimator, the ridge parameter could serve for regularization of an ill-conditioned design matrix, while the tuning parameter could be used for tuning the fit quality effectively, and as the ridge parameter increases, the Liu-Type estimator produces more robust regression models than the ridge estimator. Numerical examples are given to illustrate the theoretical results.
For linear model with stochastic linear restrictions, in this thesis we generalize the traditional ordinary mixed estimator (OME) to singular linear model and proposed singular mixed estimator (SME). Performances of the SME in the MSEM sense and its two-stage estimator are also discussed. Applying the SME in the parameter estimation of Panel data model with stochastic linear restrictions, we derived four feasible estimators for Panel data model. The relationship among them is discussed and it’s shown that they are superior to the corresponding unrestricted estimators. Furthermore, we have studied the prediction problem in singular linear model based on the SME. The optimal heterogeneous predictor, optimal unbiased homogeneous predictor and the optimal predictor for singular linear model with stochastic linear restrictions are derived, and we find that all the three predictors satisfy a general formula for prediction. By combining the idea of mixed estimation and Liu estimator, we proposed a new stochastic restricted Liu estimator and prove that it outperforms the traditional Liu estimator, OME and some other stochastic restricted estimators under certain conditions. Simulation study and numerical example have supported the theoretical results derived. For two common types of misspecification of regression models, we have further studied the behaviors of the stochastic restricted Liu estimator in such two cases, and performances of the corresponding predictors are also examined.
For the preliminary test estimator when exact linear restrictions are used, we have firstly discussed some common test methods in literatures, including the familiar F test and some large sample tests widely used in Econometric models such as the Wald test, Likelihood Ratio (LR) test and Lagrangian Multiplier (LM) test. By combing the idea of preliminary test and Liu estimator, we have proposed three preliminary test Liu estimators (PTLE) based on the Wald, LR and LM tests. Through the bias analysis, we find that the Wald test based PTLE has the smallest quadratic bias,followed by the PTLE based on the LR and LM tests. On the other hand, we find that when near the null hypothesis, the LM test based PTLE has the smallest risk, followed by the PTLE based on the LR and Wald tests, while when the parameter departs from the restrictions, the situation is reversed. In the meantime, when the Liu parameter is small, then the PTLE based on W test has the smallest MSE followed by the LR and LM tests, while when the Liu parameter is large and near one, the situation is just also reversed. Furthermore, the relative efficiency and the choice of optimal significance levels are also discussed. Considering the fat-tail phenomenon that may exist in practical data, we have also studied the PTLE for the multi-t distribution model based on the three large sample tests above. Performances of the estimators according to the quadratic bias and mean square error are similarly compared in detail.
Finally, we have also considered the preliminary test estimator when stochastic restrictions are used in regression. By imbedding the stochastic linear restriction model in an exact linear restriction framework and make use of the results concerning the quality restricted estimator, we propose the stochastic hypothesis preliminary test ridge estimator (SPTRE) based on F test. Performances of the SPTRE, stochastic hypothesis preliminary test estimator and the ridge estimator in the sense of MSE are systematically analyzed. Relative efficiency of the SPRE and the corresponding optimal choice of the significance level are also discussed in this paper.
引文
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