×

A new stochastic mixed ridge estimator in linear regression model. (English) Zbl 1247.62179

Summary: We are concerned with parameter estimation in linear regression models with additional stochastic linear restrictions. To overcome the multicollinearity problem, a new stochastic mixed ridge estimator is proposed and its efficiency is discussed. Necessary and sufficient conditions for the superiority of the stochastic mixed ridge estimator over the ridge estimator and the mixed estimator in the mean squared error matrix sense are derived for the two cases in which the parametric restrictions are correct or are not correct. Finally, a numerical example is also given to show the theoretical results.

MSC:

62J07 Ridge regression; shrinkage estimators (Lasso)
62J05 Linear regression; mixed models
62F30 Parametric inference under constraints
62H12 Estimation in multivariate analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Akdeniz F, Erol H (2003) Mean squared error matrix comparisons of some biased estimators in linear regression. Comm Stat Theory Methods 32(12): 2389–2413 · Zbl 1028.62054 · doi:10.1081/STA-120025385
[2] Baksalary JK, Trenkler G (1991) Nonnegative and positive definiteness of matrices modified by two matrices of rank one. Linear Algebra Appl 151: 169–184 · Zbl 0728.15011 · doi:10.1016/0024-3795(91)90362-Z
[3] Durbin J (1953) A note on regression when there is extraneous information about one of the coefficients. J Am Stat Assoc 48: 799–808 · Zbl 0052.15503 · doi:10.1080/01621459.1953.10501201
[4] Farebrother RW (1976) Further results on the mean square error of ridge regression. J Roy Stat Soc Ser 38(B): 248–250 · Zbl 0344.62056
[5] Gruber MHJ (1998) Improving efficiency by Shrinkage: the James-Stein and ridge regression estimators. Marcel Dekker, Inc., New York · Zbl 0920.62085
[6] Groß J (2003) Restricted ridge estimation. Stat Prob Lett 65: 57–64 · Zbl 1116.62368 · doi:10.1016/j.spl.2003.07.005
[7] Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for non-orthogonal problems. Technometrics 12: 55–67 · Zbl 0202.17205 · doi:10.1080/00401706.1970.10488634
[8] Kaciranlar S, Sakallioglus S, Akdeniz F (1998) Mean squared error comparisons of the modified ridge regression estimator and the restricted ridge regression estimator. Comm Stat Theory Methods 27(1): 131–138 · Zbl 0960.62068 · doi:10.1080/03610929808832655
[9] Kaciranlar S, Sakallioglu S, Akdeniz F, Styan GPH, Werner HJ (1999) A new biased estimator in linear regression and a detailed analysis of the widely-analysed dataset on Portland Cement. Sankhya Indian J Stat 61(B): 443–459
[10] Liu K. (1993) A new class of biased estimate in linear regression. Comm Stat Theory Methods 22: 393–402 · Zbl 0784.62065 · doi:10.1080/03610929308831027
[11] Hubert MH, Wijekoon P (2006) Improvement of the Liu estimator in the linear regression model. Stat Pap 47: 471–479 · Zbl 1125.62055 · doi:10.1007/s00362-006-0300-4
[12] Rao CR, Toutenburg H (1995) Linear models: least squares and alternatives. Springer, New York · Zbl 0846.62049
[13] Sarkar N (1992) A new estimator combining the ridge regression and the restricted least squares methods of estimation. Comm Stat Theory Methods 21: 1987–2000 · Zbl 0774.62074 · doi:10.1080/03610929208830893
[14] Swindel BF (1976) Good estimators based on prior information. Comm Stat Theory Methods 5: 1065–1075 · Zbl 0342.62035 · doi:10.1080/03610927608827423
[15] Stein C (1956) Inadmissibility of the usual estimator for mean of multivariate normal distribution. In: Neyman J (ed) Proceedings of the third berkley symposium on mathematical and statistics probability vol 1, pp 197–206 · Zbl 0073.35602
[16] Theil H, Goldberger AS (1961) On pure and mixed statistical estimation in economics. Intern Econ Rev 2: 65–78 · doi:10.2307/2525589
[17] Theil H (1963) On the use of incomplete prior information in regression analysis. J Am Sta Assoc 58: 401–414 · Zbl 0129.11401 · doi:10.1080/01621459.1963.10500854
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.