Generalized Multivariate Ridge Regression Estimate and Criteria Q(c)for Choosing Matrix K

Chen Shi-ji; Zeng Zhi-bin

Volume 14 Issue 1

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Chen Shi-ji, Zeng Zhi-bin. Generalized Multivariate Ridge Regression Estimate and Criteria Q(c)for Choosing Matrix K[J]. Applied Mathematics and Mechanics, 1993, 14(1): 69-78.

Citation:

Chen Shi-ji, Zeng Zhi-bin. Generalized Multivariate Ridge Regression Estimate and Criteria Q(c)for Choosing Matrix K[J]. Applied Mathematics and Mechanics, 1993, 14(1): 69-78.

Citation:

Chen Shi-ji, Zeng Zhi-bin. Generalized Multivariate Ridge Regression Estimate and Criteria Q(c)for Choosing Matrix K[J]. Applied Mathematics and Mechanics, 1993, 14(1): 69-78.

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Generalized Multivariate Ridge Regression Estimate and Criteria Q(c)for Choosing Matrix K

Department of Mathematics, Fujian Normal University, Fuzhou

Received Date: 1991-10-18
Publish Date: 1993-01-15

Abstract

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.
- least square estimate,
- generalized ridge estimate,
- mean square error

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References(7)

References

[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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