dest.liu {liureg}

Computation of Liu Biasing Parameter d

Description

The dest function computes different Liu biasing parameters available in the literature, proposed by different researchers such as given in Liu (1993) <doi:10.1080/03610929308831027>, Liu (2011) <doi:10.1016/j.jspi.2010.05.030>, and Ozkale and Kaciranlar (2007) <doi:10.1080/03610920601126522>.

Usage

dest(object, ...)
## S3 method for class 'liu'
dest(object, ...)
## S3 method for class 'dliu'
print(x, ...)

Arguments

object

An object of class "liu".

x

An object of class "dliu" for the print.dest.dliu.

...

Not presently used in this implementation.

Details

The dest function computes different biasing parameter for the Liu regression. All these methods are already available in the literature proposed by various authors.

Value

This function returns the list of following estimators of Liu parameter, available in the literature.

dopt

By Liu (1993): \frac{\sum_{j=1}^p \left[\frac{\alpha_j^2-\sigma^2}{(\lambda_j+1)^2}\right]}{\sum_{j=1}^p\left[\frac{\sigma^2+\lambda_j\times\alpha_j^2}{\lambda_j(\lambda_j+1)^2}\right]}

dILE

By Liu, (2011): \frac{\sum_{j=1}^n \left(\frac{\widetilde{e}_i}{1-g_{ii}}\left(\frac{\widetilde{e}_i}{1-h_{1-ii}}-\frac{\hat{e}_i}{1-h_{ii}}\right)\right)} {\sum_{i=1}^n \left(\frac{\widetilde{e}_i}{1-g_{ii}}-\frac{\hat{e}_i}{1-h_{ii}}\right)^2},

where, \hat{e}=y_i-x'_i(X'X-x_i x'_i)^{-1}(X'y-x_iy_i), \widetilde{e}=y_i-x'_i(X'X+I_p-x_i x'_i)^{-1}(X'y-x_iy_i), G=X(X'X+I_p)^{-1}X' and H\cong X(X'X)^{-1}X'.

dmm

Liu, (1993): (1-\hat{\sigma}^2) \left[\frac{\sum_{j=1}^p\left[\frac{1}{\lambda_j(\lambda_j+1)}\right]}{\sum_{j=1}^p\left[\frac{\hat{\alpha}_j^2}{(\lambda_j+1)^2}\right]}\right]

dcl

By Liu, (1993): (1-\hat{\sigma}^2)\left[\frac{\sum_{j=1}^p\left[\frac{1}{(\lambda_j+1)}\right]}{\sum_{j=1}^p \left[\frac{\lambda_j\hat{\alpha}_j^2}{(\lambda_j+1)^2}\right]}\right].

GCV

GCV criterion for selection of optimal d, that is, GCV=\frac{SSR_d}{(n-1-trace(H_d))}, where SSR_d is residuals sum of squares from Liu regression at certain value of d and trace(H_d) is trace of hat matrix from Liu.

dGCV

returns value of d at which GCV is minimum.

Author(s)

Muhammad Imdad Ullah, Muhammad Aslam

References

Akdeniz, F. and Kaciranlar, S. (1995). On the Almost Unbiased Generalized Liu Estimators and Unbiased Estimation of the Bias and MSE. Communications in Statistics-Theory and Methods, 24, 1789–1897. http://doi.org/10.1080/03610929508831585.

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

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.

Liu, K. (1993). A new Class of Biased Estimate in Linear Regression. Journal of Statistical Planning and Inference, 141, 189–196. http://doi.org/10.1080/03610929308831027.

Liu, X-Q. (2011). Improved Liu Estimator in a Linear Regression Model. Journal of Statistical Planning and Inference,141, 189–196. https://doi.org/10.1016/j.jspi.2010.05.030.

Ozkale, R. M. and Kaciranlar, S. (2007). A Prediction-Oriented Criterion for Choosing the Biasing Parameter in Liu Estimation. Commincations in Statistics-Theory and Methods, 36(10): 1889–1903. http://doi.org/10.1080/03610920601126522.

See Also

Liu model fitting liu, Liu residuals residuals.liu, Liu PRESS press.liu, Testing of Liu coefficients summary.liu

Examples

mod<-liu(y ~ ., data = as.data.frame(Hald), d = seq(-5, 5, 0.1))
dest(mod)
## Vector of GCV values for each d
dest(mod)$GCV