kest: Computation of Ridge Biasing Parameter K

In lmridge: Linear Ridge Regression with Ridge Penalty and Ridge Statistics

Computation of Ridge Biasing Parameter K

Description

The kest function computes different biasing parameters available in the literature proposed by different researchers.

Usage

kest(object, ...)
## S3 method for class 'lmridge'
kest(object, ...)
## S3 method for class 'klmridge'
print(x, ...)

Arguments

object	An object of class "lmridge" for the kest.
x	An object of class "klmridge" for the print.kest.klmridge.
...	Not presently used in this implementation.

Details

The kest function computes different biasing parameter for the ordinary linear ridge regression. All these methods are already available in the literature and proposed by various authors. See reference section.

Value

The function returns the list of following biasing parameter methods proposed by various researchers.

mHKB	By Thisted (1976), (p-2)\hat{σ}^2/{(\hat{β}'\hat{β})}
LW	As in lm.ridge of MASS (p-2)\hat{σ}^2*n/∑(\hat{y}^2)
LW76	By Lawless and Wang (1976), (p\hat{σ}^2)/(∑{λ_j*\hat{α}_j^2})
CV	Value of Cross Validation (CV) for each biasing parameter K, CV_k=1/n ∑_{j=1}^n (y_i-X_j \hat{β}_{j_K})^2.
kCV	Value of biasing parameter at which CV is small.
HKB	By Hoerl and Kennard (1970), (p\hat{σ}^2)/(\hat{β}'\hat{β})
kibAM	By Kibria (2003), 1/p*(∑{\hat{σ^2}/\hat{β}_j^2)}
GCV	Value of Generalized Cross Validation (GCV) for each biasing parameter K, (∑_{j=1}^n (y_i-\hat{β}_{j_K}))^2/[n-(1+Trace(H_{R,k}))]^2.
kcGCV	Value of biasing parameter at which GCV is small.
DSK	By Dwividi and Shrivastava, (1978), \hat{σ}^2/(\hat{β}'\hat{β})
kibGM	By Kibria (2003), (\hat{σ}^2)/(∏\hat{α}_j^2)^(1/p)
kibMEd	By Kibria (2003), median[(\hat{σ}^2)/(\hat{α}_j^2)]
KM2	By Muniz and Kibria (2009), max[(1)/(√{(\hat{σ}^2)/(\hat{α}_j^2)})]
KM3	By Muniz and Kibria (2009), max[√{(\hat{σ}^2)/(\hat{α}_j^2)}]
KM4	By Muniz and Kibria (2009), [∏ (1/(√{(\hat{σ}^2)/(\hat{α}_j^2)}))]^(1/p)
KM5	By Muniz and Kibria (2009), [∏√{(\hat{σ}^2)/(\hat{α}_j^2)}]^{1/p}
KM6	By Muniz and Kibria (2009), Median[1/√{\hat{σ}^2/\hat{α}^2_j}]
KM8	By Muniz et al. (2012), max[1/((λ_{max}\hat{σ}^2)/((n-p)\hat{σ}^2+λ_{max}\hat{α}^2_j))]
KM9	By Muniz et al. (2012), max[√{(λ_{max}\hat{σ}^2)/((n-p)\hat{σ}^2+λ_{max}\hat{α}^2_j)}]
KM10	By Muniz et al. (2012), [∏(1/√{(λ_{max}\hat{σ}^2)/((n-p)\hat{σ}^2+λ_{max}\hat{α}^2_j)}) ]^{1/p}
KM11	By Muniz et al. (2012), [∏ (√{(λ_{max}\hat{σ}^2)/((n-p)\hat{σ}^2)+λ_{max}\hat{α}^2_j})]^(1/p)
KM12	By Muniz et al., Median[(1)/(√{(λ_{max}\hat{σ}^2)/((n-p)\hat{σ}^2)+λ_{max}\hat{α}^2_j})]
KD	By Dorugade and Kashid (2012), max[0, (p\hat{σ}^2)/(\hat{α}'\hat{α})-(1)/(n(VIF_j)_{max})]
KAD4	By Dorugade and Kashid (2012), HM[2p/λ_{max}∑(\hat{σ}^2 / \hat{α}^2_j)]
alphahat	The OLS estimator in canonical form, i.e., \hat{α}=(P'X'XP)^{-1}X'^*y, where X^*=XP P is eigenvector of X'X.

Author(s)

Muhammad Imdad Ullah, Muhammad Aslam

References

Dorugade, A. and Kashid, D. (2010). Alternative Method for Choosing Ridge Parameter for Regression. Applied Mathematical Sciences, 4(9), 447-456.

Dorugade, A. (2014). New Ridge Parameters for Ridge Regression. Journal of the Association of Arab Universities for Basic and Applied Sciences, 15, 94-99. doi: 10.1016/j.jaubas.2013.03.005.

Hoerl, A. E., Kennard, R. W., and Baldwin, K. F. (1975). Ridge Regression: Some Simulation. Communication in Statistics, 4, 105-123. doi: 10.1080/03610927508827232.

Hoerl, A. E. and Kennard, R. W., (1970). Ridge Regression: Biased Estimation of Nonorthogonal Problems. Technometrics, 12, 55-67. doi: 10.1080/00401706.1970.10488634.

Imdad, M. U. Addressing Linear Regression Models with Correlated Regressors: Some Package Development in R (Doctoral Thesis, Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan), 2017.

Kibria, B. (2003). Performance of Some New Ridge Regression Estimators. Communications in Statistics-Simulation and Computation, 32(2), 491-435. doi: 10.1081/SAC-120017499.

Lawless, J., and Wang, P. (1976). A Simulation Study of Ridge and Other Regression Estimators. Communications in Statistics-Theory and Methods, 5(4), 307-323. doi: 10.1080/03610927608827353.

Muniz, G., and Kibria, B. (2009). On Some Ridge Regression Estimators: An Empirical Comparisons. Communications in Statistics-Simulation and Computation, 38(3), 621-630. doi: 10.1080/03610910802592838.

Muniz, G., Kibria, B., Mansson, K., and Shukur, G. (2012). On developing Ridge Regression Parameters: A Graphical Investigation. SORT-Statistics and Operations Research Transactions, 36(2), 115–138.

Thisted, R. A. (1976). Ridge Regression, Minimax Estimation and Empirical Bayes Methods. Technical Report 28, Division of Biostatistics, Stanford University, California.

Venables, W. N. and Ripley, B. D. (2002). Modern Applied Statistics with S. Springer New York, 4th edition, ISBN 0-387-95457-0.

See Also

The ridge model fitting lmridge, Ridge Var-Cov matrix vcov

Examples

mod <- lmridge(y~., as.data.frame(Hald), K = seq(0, 0.2, 0.001))
kest(mod)

## GCV values
kest(mod)$GCV

## minimum GCV value at certain k
kest(mod)$kGCV

## CV Values
kest(mod)$CV

## minimum CV value at certain k
kest(mod)$kCV

## Hoerl and Kennard (1970)
kest(mod)$HKB

lmridge documentation built on Jan. 15, 2023, 5:06 p.m.