多元广义岭估计及K值选取的Q(c)准则*
Generalized Multivariate Ridge Regression Estimate and Criteria Q(c)for Choosing Matrix K
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摘要: 当自变量间存在复共线性时,最小二乘估计就表现出不稳定并可能导致错误的结果.本文采用广义岭估计β(K)来估计多元线性模型的回归系数β=vec(B),通过岭参数K值的选取,可使广义岭估计的均方误差MSE小于最小二乘估计的MSE.指出了广义岭估计中根据MSE准则选取K值存在的主要缺陷,采用了一种选取K值的新准则Q(c),它包含MSE准则和最小二乘LS准则作为特例,从理论上证明和讨论了Q(c)准则的优良性,阐明了c值的统计含义,并给出了确定c值的方法.Abstract: When multicollinearity is present in a set of the regression variables, the least square estimate of the regression coefficient tends to be unstable and it may lead to erroneous inference.In this paper, generalized ridge estimate β*(K) of the regression coefficient β=vec(B) is considered in multivaiale linear regression model. The MSE of the above estimate is less than the MSE of the least square estimate by choosing the ridge parameter matrix K. Moreover, it is pointed out that the Criterion MSE for choosing matrix K of generalized ridge estimate has several weaknesses. In order to overcome these weaknesses, a new family of criteria Q(c) is adpoted which includes the criterion MSE and criterion LS as its special case. The good properties of the criteria Q(c) are proved and discussed from theoretical point of view. The statistical meaning of the scale c is explained and the methods of determining c are also given.
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Key words:
- least square estimate /
- generalized ridge estimate /
- mean square error
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[1] 陈希孺、王松桂,《近代回归分析—原理、方法及应用》.安徽教育出版社(1987). [2] 王松桂.《线性模型的理论及其应用》.安徽教育出版社(1987). [3] Brown,P,J,and J,V,Zidek,Adaptive multivariate ridge regression,Ann,Statist,,8(1)(1980),64-74. [4] Haitovkp,Yoel,On multivariate ridge regression,Biometrika,74(3)(1987),583-570. [5] Hemmerle,W.J,and T.F.Brantle,Explicit and constrained generalized ridge estimation,Technometrics,20 (1978),109-120. [6] Wichem,D,W.and G.A,Churchill,A comparison of ridge estimators,Technometrics,20(1978),301-311. [7] 鲁国斌.广义岭回归估计中关于K值选取的Q(c)准则,数理统计与应用概率.4(2)(1989),159-169. -
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