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Mean squared error matrix comparisons of some biased estimators in linear regression. (English) Zbl 1028.62054

Summary: Consider the linear regression model \(y=X\beta+u\) in the usual notation. In the presence of multicollinearity certain biased estimators like the ordinary ridge regression estimator \(\widehat \beta_d=(X'X+I)^{-1}X'y\) and the Liu estimator \(\widehat\beta_d=(X'X+I)^{-1}(X'y+d\widehat\beta)\) introduced by K. Liu [Commun. Stat., Theory Methods 22, 393-402 (1993; Zbl 0784.62065)], or improved ridge and Liu estimators are used to outperform the ordinary least squares estimates in the linear regression model. We compare the (almost unbiased) generalized ridge regression estimator with the (almost unbiased) generalized Liu estimator in the matrix mean square error sense.

MSC:

62J07 Ridge regression; shrinkage estimators (Lasso)
62H12 Estimation in multivariate analysis
62J05 Linear regression; mixed models

Citations:

Zbl 0784.62065
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References:

[1] Akdeniz F., Journal of Applied Statistical Science 9 pp 73– (1999)
[2] DOI: 10.1080/03610929508831585 · Zbl 0937.62612 · doi:10.1080/03610929508831585
[3] Farebrother R. W., Journal of the Royal Statistical Society B. 38 pp 248– (1976)
[4] DOI: 10.1080/03610929908832353 · Zbl 0919.62069 · doi:10.1080/03610929908832353
[5] DOI: 10.1006/jmva.1995.1047 · Zbl 0844.62052 · doi:10.1006/jmva.1995.1047
[6] Gruber M. H. J., Improving Efficiency by Shrinkage: The James Stein and Ridge Regression Estimators (1998) · Zbl 0920.62085
[7] DOI: 10.2307/1268701 · Zbl 0399.62071 · doi:10.2307/1268701
[8] DOI: 10.2307/1268658 · Zbl 0346.62046 · doi:10.2307/1268658
[9] DOI: 10.2307/1267351 · Zbl 0202.17205 · doi:10.2307/1267351
[10] DOI: 10.2307/1267352 · Zbl 0202.17206 · doi:10.2307/1267352
[11] DOI: 10.1080/03610927508827232 · Zbl 0296.62062 · doi:10.1080/03610927508827232
[12] Kaçiranlar S., Sankhya: The Indian Journal of Statistics 61 pp 443– (1999)
[13] Lawless J. F., Communications in Statistics–Theory and Methods 5 pp 307– (1976)
[14] DOI: 10.1080/03610929308831027 · Zbl 0784.62065 · doi:10.1080/03610929308831027
[15] DOI: 10.2307/2285832 · Zbl 0319.62049 · doi:10.2307/2285832
[16] DOI: 10.1080/03610918808812690 · Zbl 0695.62174 · doi:10.1080/03610918808812690
[17] Rao C. R., Linear Models Least Squares and Alternatives (1995) · Zbl 0846.62049 · doi:10.1007/978-1-4899-0024-1
[18] DOI: 10.1081/STA-100002036 · Zbl 1009.62559 · doi:10.1081/STA-100002036
[19] Singh B., Sankhya : The Indian Journal of Statistics 48 pp 342– (1986)
[20] Stein, C. Inadmissibility of usual estimator for the mean of a multi-variate normal distribution. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. pp.197–206. Berkeley: University of California Press.
[21] Theobald C. M., Journal of the Royal Statistical Society B. 36 pp 103– (1974)
[22] DOI: 10.1007/BF02924687 · Zbl 0703.62066 · doi:10.1007/BF02924687
[23] Troskie C. G., South African Statistical Journal 30 pp 119– (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.