| lamest {fastliu} | R Documentation |
Computation of Liu Tuning Parameter
Description
lamest computes the Liu tuning parameters provided in the literature.
The tuning parameter estimates are based on
Liu (1993) doi:10.1080/03610929308831027,
Ozkale and Kaciranlar (2007) doi:10.1080/03610920601126522,
Liu (2011) doi:10.1016/j.jspi.2010.05.030.
Usage
lamest(obj, ...)
Arguments
obj |
An object of class |
... |
Not used in this implemetation. |
Details
The lamest function computes the following tuning
parameter estimates available in the literature.
lam.mm (Liu, 1993) | \displaystyle{1-\hat{\sigma}^{2}\left(\frac{\sum\limits _{j=1}^{p}\frac{1}{\lambda_{j}\left(1+\lambda_{j}\right)}}{\sum\limits _{j=1}^{p}\frac{\hat{\alpha}_{j}^{2}}{\left(1+\lambda_{j}\right)^{2}}}\right)} |
lam.CL (Liu, 1993) | \displaystyle{1-\hat{\sigma}^{2}\left(\frac{\sum\limits _{j=1}^{p}\frac{1}{\left(1+\lambda_{j}\right)}}{\sum\limits _{j=1}^{p}\frac{\lambda_{j}\hat{\alpha}_{j}^{2}}{\left(1+\lambda_{j}\right)^{2}}}\right)} |
lam.opt (Liu, 1993) | \displaystyle{\frac{\sum\limits _{j=1}^{p}\left(\frac{\alpha_{j}^{2}-\sigma^{2}}{\left(1+\lambda_{j}\right)^{2}}\right)}{\sum\limits _{j=1}^{p}\left(\frac{\sigma^{2}+\lambda_{j}\alpha_{j}^{2}}{\lambda_{j}\left(1+\lambda_{j}\right)^{2}}\right)}} |
lam.OK (Ozkale and Kaciranlar, 2007; Liu, 2011) | \displaystyle{\frac{\sum\limits _{i=1}^{n}\frac{\tilde{e}_{i}}{1-g_{ii}}\left(\frac{\tilde{e}_{i}}{1-h_{1-ii}}-\frac{\hat{e}_{i}}{1-h_{ii}}\right)}{\sum\limits _{i=1}^{n}\left(\frac{\tilde{e}_{i}}{1-g_{ii}}-\frac{\hat{e}_{i}}{1-h_{ii}}\right)^{2}}} with \hat{e}_{i}=y_{i}-\mathbf{x}_{i}^{T}\left(\mathbf{X}^{T}\mathbf{X}-\mathbf{x}_{i}\mathbf{x}_{i}^{T}\right)^{-1}\left(\mathbf{X}^{T}\mathbf{y}-\mathbf{x}_{i}y_{i}\right) and \tilde{e}_{i}=y_{i}-\mathbf{x}_{i}^{T}\left(\mathbf{X}^{T}\mathbf{X}+\mathbf{I}-\mathbf{x}_{i}\mathbf{x}_{i}^{T}\right)^{-1}\left(\mathbf{X}^{T}\mathbf{y}-\mathbf{x}_{i}y_{i}\right) where g_{ii} and h_{ii} are the ith diagonal elements of \mathbf{G}=\mathbf{X}\left(\mathbf{X}^{T}\mathbf{X}+\mathbf{I}\right)^{-1}\mathbf{X}^{T} and \mathbf{H=}\mathbf{X}\left(\mathbf{X}^{T}\mathbf{X}\right)^{-1}\mathbf{X}^{T}, respectively. |
lam.GCV | This is the \lambda value corresponding to the minimum of the generalized cross-validition (GCV) values. The GCV is computed by \frac{\mathrm{SSR}_{\lambda}}{n-1-\mathrm{trace}\left(\mathbf{H}_{\lambda}\right)} where \mathrm{SSR}_{\lambda} is the residual sum of squares and \mathrm{trace}\left(\mathbf{H}_{\lambda}\right) is the trace of the hat matrix at corresponding value of \lambda from Liu regression. |
Value
The return object is the Liu tuning parameter estimates based on the literature.
Author(s)
Murat Genç and Ömer Özbilen
References
Liu, K. (1993). A new class of blased estimate in linear regression. Communications in Statistics-Theory and Methods, 22(2), 393-402. doi:10.1080/03610929308831027.
Liu, X. Q. (2011). Improved Liu estimator in a linear regression model. Journal of Statistical Planning and Inference, 141(1), 189-196. doi:10.1016/j.jspi.2010.05.030.
Ozkale, M. R. and Kaciranlar, S. (2007). A prediction-oriented criterion for choosing the biasing parameter in Liu estimation. Communications in Statistics-Theory and Methods, 36(10), 1889-1903. doi:10.1080/03610920601126522. Imdadullah, M., Aslam, M., and Altaf, S., (2017). liureg: A Comprehensive R Package for the Liu Estimation of Linear Regression Model with Collinear Regressors. The R Journal, 9(2), 232-247.
See Also
liureg(), predict(), summary(), pressliu(), residuals()
Examples
Hitters <- na.omit(Hitters)
X <- model.matrix(Salary ~ ., Hitters)[, -1]
y <- Hitters$Salary
lam <- seq(0, 1, 0.01)
liu.mod <- liureg(X, y, lam)
lamest(liu.mod)