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.
lamest(obj, ...)The return object is the Liu tuning parameter estimates based on the literature.
An object of class liureg.
Not used in this implemetation.
Murat Genç and Ömer Özbilen
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 \(i\)th 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. |
Liu, K. (1993). A new class of blased estimate in linear regression. Communications in Statistics-Theory and Methods, 22(2), 393-402. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_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.
liureg(), predict(), summary(), pressliu(), residuals()
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)
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