Further Results on the Mean Square Error of Ridge Regression

Farebrother, R. W.

doi:10.1111/j.2517-6161.1976.tb01588.x

Cited by 277 publications

(76 citation statements)

References 4 publications

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“…The following notations and lemmas are needful to prove the statistical property of

{\overset{truê}{α}}_{MRT} ((), k, d) .

Lemma Let n × n matrices M > 0, N > 0 (or N ≥ 0), then M > N if and only if λ 1 (NM ‐1 ) < 1, where λ 1 (NM ‐1 ) is the largest eigenvalue of the matrix NM ‐1 .Lemma Let M be an n × n positive definite matrix, that is M > 0, and α be some vector, then M − αα ′ ≥ 0 if and only if α ′ M −1 α ≤ 1.Lemma Let

{\overset{truê}{α}}_{i} = A_{i} y i = 1, 2

be two linear estimators of α . Suppose that

D = italicCov ((), {\overset{α}{true}}_{1}) - italicCov ((), {\overset{α}{true}}_{2}) > 0

, where

italicCov ((), {\overset{α}{true}}_{normali}), i = 1, italic2

denotes the covariance matrix of

{\overset{truê}{α}}_{i}

and

b_{i} = italicBias ((), {\overset{α}{true}}_{normali}) = ((), A_{i} X - I) α, i = 1, 2

.…”

Section: Some Existing Estimators and The Mrt Estimatormentioning

confidence: 99%

Modified ridge‐type estimator to combat multicollinearity: Application to chemical data

Lukman

Ayinde

Binuomote

et al. 2019

Journal of Chemometrics

111

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The Linear regression model is one of the most widely used models in different fields of study. The most popularly used estimation technique is the ordinary least squares estimator. The technique becomes unstable and gives misleading result in the presence of multicollinearity. The ridge regression estimator has been widely accepted as an alternative method to combat the problem of multicollinearity. In this study, a modified ridge‐type estimator is suggested by modifying the ridge regression estimator. A Monte Carlo simulation study and real‐life application were conducted to compare the performance of this estimator and some other existing estimators. The results of both simulation study and real‐life application show that the proposed estimator outperforms other competing estimator.

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“…The following notations and lemmas are needful to prove the statistical property of

{\overset{truê}{α}}_{MRT} ((), k, d) .

{\overset{truê}{α}}_{i} = A_{i} y i = 1, 2

be two linear estimators of α . Suppose that

D = italicCov ((), {\overset{α}{true}}_{1}) - italicCov ((), {\overset{α}{true}}_{2}) > 0

, where

italicCov ((), {\overset{α}{true}}_{normali}), i = 1, italic2

denotes the covariance matrix of

{\overset{truê}{α}}_{i}

and

b_{i} = italicBias ((), {\overset{α}{true}}_{normali}) = ((), A_{i} X - I) α, i = 1, 2

.…”

Section: Some Existing Estimators and The Mrt Estimatormentioning

confidence: 99%

Modified ridge‐type estimator to combat multicollinearity: Application to chemical data

Lukman

Ayinde

Binuomote

et al. 2019

Journal of Chemometrics

111

View full text Add to dashboard Cite

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“…[36] for a similar but more statistically oriente d proof. If such a matrix is subtracted from a positive definite matrix , then the resulting matrix is not necessarily nonnegative definite.…”

Section: Rank 1 Modificationmentioning

confidence: 76%

Linear Regression

Groß

2003

Lecture Notes in Statistics

101

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“…This condition is also the condition for the RR estimator to be better than the OLS estimator by the matrix MSE criterion (see also [19] and [18]). Therefore, if there exists a k value for the estimatorβ m (k) to be superior to the estimatorβ m , the estimatorβ(k) is superior to the estimatorβ L for the same k in the sense of the matrix MSE criterion and the converse is also true.…”

Section: Lemma Letmentioning

confidence: 89%

A stochastic restricted ridge regression estimator

Özkale

2009

Journal of Multivariate Analysis

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a b s t r a c t Groß [J. Groß, Restricted ridge estimation, Statistics & Probability Letters 65 (2003) 57-64] proposed a restricted ridge regression estimator when exact restrictions are assumed to hold. When there are stochastic linear restrictions on the parameter vector, we introduce a new estimator by combining ideas underlying the mixed and the ridge regression estimators under the assumption that the errors are not independent and identically distributed. Apart from [J. Groß, Restricted ridge estimation, Statistics & Probability Letters 65 (2003) 57-64], we call this new estimator as the stochastic restricted ridge regression (SRRR) estimator. The performance of the SRRR estimator over the mixed estimator in respect of the variance and the mean square error matrices is examined. We also illustrate our findings with a numerical example. The shrinkage generalized least squares (GLS) and the stochastic restricted shrinkage GLS estimators are proposed.

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Further Results on the Mean Square Error of Ridge Regression

Abstract: CONSIDER the model y = X(3+e, Be = 0, Eee T = 0' 2 1,

Cited by 277 publications

References 4 publications

Modified ridge‐type estimator to combat multicollinearity: Application to chemical data

Modified ridge‐type estimator to combat multicollinearity: Application to chemical data

Linear Regression

A stochastic restricted ridge regression estimator

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