关于有偏估计若干问题的进一步研究
|
摘要
线性模型是很重要的一类统计模型，它包括线性回归模型、方差分析模型、协方差分析模型和方差分量模型等等。论文主要针对一般线性回归模型和广义线性回归模型，即：，和，其中，为向量，为设计矩阵，且，为向量，为向量，是已知的正定阵。由于是未知参数，因此研究参数及其线性函数的估计极其重要。论文基于最小二乘估计及有偏估计特别是岭估计，对参数的约束条件做了进一步研究，并提出一种新型估计即广义岭型估计；对模型的点预测问题进行深入探索，得出一种基于岭估计关于经典预测和最优预测的最优性判别条件；也对回归诊断特别是基于主成分估计的距离进行了深入探讨。主要结果如下：
论文第三章从设计矩阵的多重共线性角度出发，考虑回归系数的椭球约束，获得了椭球约束下线性模型参数的一种新型估计--广义岭型估计。该估计虽然具有偏崎，但其估计精度具有良好的性质，如：有偏性、方差一致最优性、相对于广义最小二乘估计的广义方差效率、MDE——有效性等。
第四章以岭估计为基础，以平均离差矩阵为判别准则，对线性模型的最优预测量与经典预测量的最优性判别问题进行了讨论。借助矩阵中L?wner偏序的一些性质，获得在此判别准则下判别两类预测量最优性的充要条件。为研究基于有偏估计关于两类预测量的最优性判别问题提供了一种方法和思路。
针对设计矩阵的多重共线性问题，为了改进基于最小二乘估计的统计诊断量Cook距离，提出了基于Massy主成分下的Cook距离(MPCC距离)。采用删除数据法，得出MPCC距离关于杠杆值和残差的精确简化式。这既大大简化了运算过程，也为实际应用中判断强影响数据提供了一种诊断方法。
Linear models are especially important statistical models, including linear regression model, variance and analysis, covariance and analysis, and variance and component one etc.. This thesis mainly considers ordinary linear regression model and generalized one, that is, models and are involved. In term of the unknown parameter , it is necessary to study its estimation. Based on the least squares and biased estimation especially ridge estimation, a new estimation, that is, generalized ridge estimation is put forward through studies on restriction of the parameter . Model's prediction being considered, comparison of superiority of optimal and classical predictions with respect to the ridge estimation is showed. Regression diagnoses especially distance for principal components estimation is discussed. The main results are as follows:
According to the approximate multicollinearity of matrix , the third chapter constrains the regression coefficient and obtains generalized ridge estimation of the linear model's parameter under the ellipsoidal restriction. Then discusses its properties, such as biased property, relative efficiency of generalized variance and superiority comparisons between generalized ridge estimation and generalized least squares estimation. Shows iterative algorithm based on the mean dispersion error.
In term of the prediction problem, the fourth chapter discusses its superiority of the optimal and classical predictors based on the ridge estimation, and gives an necessary and sufficient condition of comparison of its superiority under the condition of criterion by some properties of partial ordering of matrix. Thus proposes an alternative method for the research of superiority of two predictors based on the biased estimation.
In the light of the approximate multicollinearity of matrix, distance for principal components estimation (namely distance) is put forward. Deletion is employed and the exact deletion formula for distance is gained, which not only simplifies the computation but also proposes a diagnostic for identifying influential observations.
论文第三章从设计矩阵的多重共线性角度出发，考虑回归系数的椭球约束，获得了椭球约束下线性模型参数的一种新型估计--广义岭型估计。该估计虽然具有偏崎，但其估计精度具有良好的性质，如：有偏性、方差一致最优性、相对于广义最小二乘估计的广义方差效率、MDE——有效性等。
第四章以岭估计为基础，以平均离差矩阵为判别准则，对线性模型的最优预测量与经典预测量的最优性判别问题进行了讨论。借助矩阵中L?wner偏序的一些性质，获得在此判别准则下判别两类预测量最优性的充要条件。为研究基于有偏估计关于两类预测量的最优性判别问题提供了一种方法和思路。
针对设计矩阵的多重共线性问题，为了改进基于最小二乘估计的统计诊断量Cook距离，提出了基于Massy主成分下的Cook距离(MPCC距离)。采用删除数据法，得出MPCC距离关于杠杆值和残差的精确简化式。这既大大简化了运算过程，也为实际应用中判断强影响数据提供了一种诊断方法。
Linear models are especially important statistical models, including linear regression model, variance and analysis, covariance and analysis, and variance and component one etc.. This thesis mainly considers ordinary linear regression model and generalized one, that is, models and are involved. In term of the unknown parameter , it is necessary to study its estimation. Based on the least squares and biased estimation especially ridge estimation, a new estimation, that is, generalized ridge estimation is put forward through studies on restriction of the parameter . Model's prediction being considered, comparison of superiority of optimal and classical predictions with respect to the ridge estimation is showed. Regression diagnoses especially distance for principal components estimation is discussed. The main results are as follows:
According to the approximate multicollinearity of matrix , the third chapter constrains the regression coefficient and obtains generalized ridge estimation of the linear model's parameter under the ellipsoidal restriction. Then discusses its properties, such as biased property, relative efficiency of generalized variance and superiority comparisons between generalized ridge estimation and generalized least squares estimation. Shows iterative algorithm based on the mean dispersion error.
In term of the prediction problem, the fourth chapter discusses its superiority of the optimal and classical predictors based on the ridge estimation, and gives an necessary and sufficient condition of comparison of its superiority under the condition of criterion by some properties of partial ordering of matrix. Thus proposes an alternative method for the research of superiority of two predictors based on the biased estimation.
In the light of the approximate multicollinearity of matrix, distance for principal components estimation (namely distance) is put forward. Deletion is employed and the exact deletion formula for distance is gained, which not only simplifies the computation but also proposes a diagnostic for identifying influential observations.
引文
[1] Rao. C. R. Linear Statistical Inferences and Its Applications.Wiley New York.1973
[2] 杨虎. 单参数主成分回归估计. 高校应用数学学报.1989 .4(1).74-80
[3] 杨虎. Gauss-Markov模型存在多重共线性时的一般主成分回归. 重庆交通学院学报. 1988.
26（3）109-115
[4] 贾忠贞. 回归系数的组合主成分估计. 数理统计与应用概率.1987.2（2）.153-158
[5] 夏结来. 回归系数的广义根方估计. 数理统计与应用概率.1989.7（4）.29-33
[6] 夏结来. 回归系数的根方有偏估计及其应用. 数理统计与应用概率.1988.3（3）.21-30
[7] 林路. 回归系数的综合岭估计. 数理统计与应用概率. 1996.11（3）.180-184
[8] Robert H. Crouse & Chun Jin. Unbiased Ridged Estimation with Prior Information and Ridge
Trace. Commun. Statist. 1995. 24（9）. 2341-2354
[9] 陈立萍. 一种岭回归估计的计算方法. 数理统计与应用概率. 1992. 8（1）.38-44
[10] 何中市,何良材. 岭回归估计值选取迭代算法的收敛性定理和极限. 应用数学学报.
1994. 17（1）. 60-64
[11] 汪明瑾，王静龙. 岭回归中确定值的一种方法. 数理统计与应用概率. 2001. 1（2）.7-14
[12] 鲁国斌. 广义岭回归估计中关于值选取的Q(c)准则. 数理统计与应用概率.1995.
4（2）.159-169
[13] Smith, G. and Campbell, F., J. Amer. Statist. Assoc.. 1980. 75. 74-100
[14] Draper, N. R. and Nostrand, R, C. V.. Technometrics. 1979. 27. 451-66
[15] Kaciranlar S.,Akdeniz F.. Mean Squared Error Comparisons of the Modified Ridge Regression
Estimation and the Restricted Ridge Regression Estimation. Commun. Statist.. 1998. 1. 45-66
[16] 杨虎. 关于回归系数的泛岭估计类. 重庆交通学院学报. 1991. 3 (10). 42-48
[17] 戴俭华，王石清. 岭估计优于最小二乘估计的条件. 数理统计与应用概率.1994. 9（2）
54-58
[18] 王惠文，朱韵华. PLS回归在消除多重共线性中的作用. 数理统计与管理. 1996. 15（6）
48-52
[19] 阎慈林. 关于主成分分析做综合评价的若干问题. 数理统计与管理. 1998. 17（2）. 22-25
[20] 黄宁. 关于主成分分析应用的思考.数理统计与管理. 1999. 18（5）.44-46
[21] 马智明, 阳宪惠. PLSR模型的回归效果分析. 数理统计与管理. 2000. 19（3）.44-47
[22] Bibby, J. and Toutenburg, H.. Prediction and Imprved Estimation in Linear Models. New York.
1977
[23] Rao,C.R, H.Toutenberg. Linear Models. Springer-Verlag. New York, Berlin, Heidelberg. 1995
[24] 王松桂. 回归诊断发展综述 应用概率统计 1988. 4（3）. 314-319
[25] 王松桂. 线性回归诊断. 数理统计与管理. 1985. 6. 62-81. 1986. 1. 1-14
[26] Ascombe, F. J.. Examination of Residuals. Proc.Fourth. Berkeley Symp. 1961. 1. 1-36
[27] Ascombe , F. J. and Turkey, J. W.. The Examination and Analysis of Residuals,
Technometrics. 1963(5), 141-160
[28] Box, G. E. P., Cox, D. R., An Analysis of Transformations (with discussion), J. Roy. Statistic.
Soc., B. 26(1964), 211-246
[29] Caroll, R., Ruppert, D.. On Prediction And The Power Transformation Family,
Biometrika,.1981. 68. 609-616
[30] Chatterjee and A.S.Hadi., Sentivity Analysis in Linear Regression. Wiley. New York. 1988
[31] Cook，R.D.,Weisberg.S.. Residuals and Influence in Regression, Chapman and Hall, New
York. 1988
[32] 胡跃清. 基于复相关系数的回归诊断方法. 数理统计与应用概率. 1993. 8（3）17-23
[33] Bruce E.Barret and Robert F.Ling. General Classes of Influence Measures for Multivariate
Regression. J.Amer.Statist. Assoc. .1992. 87(417). 185-191
[34] 王松桂. 线性模型理论与方法的若干新近展. 数理统计与应用概率. 1987. 2（1）. 95-109
[35] 杨虎. 强影响点的一种新度量. 数理统计与应用概率. 1990. 6（2）. 212-218
[36] 赵进文. 多重共线性影响点的主成分诊断. 数理统计与应用概率. 1995. 10（2）.23-27
[37] Choogak KIM, Korea & Woochul KIM. Some Diagnostic Results in Nonparametric Density
Estimation. Commun. Statist.. 1998. 27（2）. 291-303
[38] Wesiberg, S.. Some Principles for Regression Diagnostics and Influence Analysis.
Technometrics. 1983. 25. 452-572
[39] 王松桂. 线性模型的理论及其应用. 合肥. 安徽教育出版社. 1987
[40] 王松桂. 贾忠贞. 矩阵论中不等式. 合肥. 安徽教育出版社.1994
[41] Jerzy K. Baksalary, Bernhard Schipp and G?tz Trenkler, Some Further Results on
Hermitian-Matrix Inequalities. Linear Algebra and its Applications. 1992. 160. 119-129
[42] Belsley, D. A.,Kuh. E. and Welsch. R. E.. Regression Diagnostics. New York. John, Wiley. 1980
[43] 王松桂. 线性模型参数估计的新进展. 数学进展.1985. 14(3). 194-203
[44] Cook, R.D and Weisberg .S.. Characterizations of An Empirical Influence Function
for Detecting Influential Cases in Regression. Technometrics. 1980. 22(4). 495-507
[45] Cook, R.D.. Detection of Influential in Linear Regression. Technometrics.1977. 19(1). 15-18
[46] Gunst, R. F. and Mason, R. L.. Regression Analysis and Its Applications, New York, Marcel
Dekleer.. 1980
[2] 杨虎. 单参数主成分回归估计. 高校应用数学学报.1989 .4(1).74-80
[3] 杨虎. Gauss-Markov模型存在多重共线性时的一般主成分回归. 重庆交通学院学报. 1988.
26（3）109-115
[4] 贾忠贞. 回归系数的组合主成分估计. 数理统计与应用概率.1987.2（2）.153-158
[5] 夏结来. 回归系数的广义根方估计. 数理统计与应用概率.1989.7（4）.29-33
[6] 夏结来. 回归系数的根方有偏估计及其应用. 数理统计与应用概率.1988.3（3）.21-30
[7] 林路. 回归系数的综合岭估计. 数理统计与应用概率. 1996.11（3）.180-184
[8] Robert H. Crouse & Chun Jin. Unbiased Ridged Estimation with Prior Information and Ridge
Trace. Commun. Statist. 1995. 24（9）. 2341-2354
[9] 陈立萍. 一种岭回归估计的计算方法. 数理统计与应用概率. 1992. 8（1）.38-44
[10] 何中市,何良材. 岭回归估计值选取迭代算法的收敛性定理和极限. 应用数学学报.
1994. 17（1）. 60-64
[11] 汪明瑾，王静龙. 岭回归中确定值的一种方法. 数理统计与应用概率. 2001. 1（2）.7-14
[12] 鲁国斌. 广义岭回归估计中关于值选取的Q(c)准则. 数理统计与应用概率.1995.
4（2）.159-169
[13] Smith, G. and Campbell, F., J. Amer. Statist. Assoc.. 1980. 75. 74-100
[14] Draper, N. R. and Nostrand, R, C. V.. Technometrics. 1979. 27. 451-66
[15] Kaciranlar S.,Akdeniz F.. Mean Squared Error Comparisons of the Modified Ridge Regression
Estimation and the Restricted Ridge Regression Estimation. Commun. Statist.. 1998. 1. 45-66
[16] 杨虎. 关于回归系数的泛岭估计类. 重庆交通学院学报. 1991. 3 (10). 42-48
[17] 戴俭华，王石清. 岭估计优于最小二乘估计的条件. 数理统计与应用概率.1994. 9（2）
54-58
[18] 王惠文，朱韵华. PLS回归在消除多重共线性中的作用. 数理统计与管理. 1996. 15（6）
48-52
[19] 阎慈林. 关于主成分分析做综合评价的若干问题. 数理统计与管理. 1998. 17（2）. 22-25
[20] 黄宁. 关于主成分分析应用的思考.数理统计与管理. 1999. 18（5）.44-46
[21] 马智明, 阳宪惠. PLSR模型的回归效果分析. 数理统计与管理. 2000. 19（3）.44-47
[22] Bibby, J. and Toutenburg, H.. Prediction and Imprved Estimation in Linear Models. New York.
1977
[23] Rao,C.R, H.Toutenberg. Linear Models. Springer-Verlag. New York, Berlin, Heidelberg. 1995
[24] 王松桂. 回归诊断发展综述 应用概率统计 1988. 4（3）. 314-319
[25] 王松桂. 线性回归诊断. 数理统计与管理. 1985. 6. 62-81. 1986. 1. 1-14
[26] Ascombe, F. J.. Examination of Residuals. Proc.Fourth. Berkeley Symp. 1961. 1. 1-36
[27] Ascombe , F. J. and Turkey, J. W.. The Examination and Analysis of Residuals,
Technometrics. 1963(5), 141-160
[28] Box, G. E. P., Cox, D. R., An Analysis of Transformations (with discussion), J. Roy. Statistic.
Soc., B. 26(1964), 211-246
[29] Caroll, R., Ruppert, D.. On Prediction And The Power Transformation Family,
Biometrika,.1981. 68. 609-616
[30] Chatterjee and A.S.Hadi., Sentivity Analysis in Linear Regression. Wiley. New York. 1988
[31] Cook，R.D.,Weisberg.S.. Residuals and Influence in Regression, Chapman and Hall, New
York. 1988
[32] 胡跃清. 基于复相关系数的回归诊断方法. 数理统计与应用概率. 1993. 8（3）17-23
[33] Bruce E.Barret and Robert F.Ling. General Classes of Influence Measures for Multivariate
Regression. J.Amer.Statist. Assoc. .1992. 87(417). 185-191
[34] 王松桂. 线性模型理论与方法的若干新近展. 数理统计与应用概率. 1987. 2（1）. 95-109
[35] 杨虎. 强影响点的一种新度量. 数理统计与应用概率. 1990. 6（2）. 212-218
[36] 赵进文. 多重共线性影响点的主成分诊断. 数理统计与应用概率. 1995. 10（2）.23-27
[37] Choogak KIM, Korea & Woochul KIM. Some Diagnostic Results in Nonparametric Density
Estimation. Commun. Statist.. 1998. 27（2）. 291-303
[38] Wesiberg, S.. Some Principles for Regression Diagnostics and Influence Analysis.
Technometrics. 1983. 25. 452-572
[39] 王松桂. 线性模型的理论及其应用. 合肥. 安徽教育出版社. 1987
[40] 王松桂. 贾忠贞. 矩阵论中不等式. 合肥. 安徽教育出版社.1994
[41] Jerzy K. Baksalary, Bernhard Schipp and G?tz Trenkler, Some Further Results on
Hermitian-Matrix Inequalities. Linear Algebra and its Applications. 1992. 160. 119-129
[42] Belsley, D. A.,Kuh. E. and Welsch. R. E.. Regression Diagnostics. New York. John, Wiley. 1980
[43] 王松桂. 线性模型参数估计的新进展. 数学进展.1985. 14(3). 194-203
[44] Cook, R.D and Weisberg .S.. Characterizations of An Empirical Influence Function
for Detecting Influential Cases in Regression. Technometrics. 1980. 22(4). 495-507
[45] Cook, R.D.. Detection of Influential in Linear Regression. Technometrics.1977. 19(1). 15-18
[46] Gunst, R. F. and Mason, R. L.. Regression Analysis and Its Applications, New York, Marcel
Dekleer.. 1980