g.ridge {g.ridge}

g.ridge (generalized ridge regression)

Description

Generalized ridge regression with the optimal shrinkage parameter. Ridge regression (Hoerl and Kennard, 1970) and generalized ridge regression (Yang and Emura 2017) are implemented. Tuning parameters are optimized by minimizing the CGV function (by the function CGV(.)): See Golub et al. (1979), and Sections 2.3 and 3.3 of Yang and Emura (2017).

Usage

g.ridge(X, Y, method = "HK", kmax = 500)

Arguments

X

design matrix of explanatory variables (regressors)

Y

vector of response variables

method

"HK" or "YE" for Hoerl and Kennard (1970) or Yang and Emura (2017)

kmax

maximum possible value for the shrinkage parameter (the "lambda" parameter), where the parameter is optimized in the interval (0, kmax).

Value

lambda: optimized shrinkage parameter

delta: the optimized thresholding parameter

estimate: regression coefficients (beta)

SE: Standard Error

Z: Z-value for testing beta=0

SE: P-value for testing beta=0

Sigma: variance estimate of the error distribution (the square of the standard deviation)

delta: thresholding parameter

References

Yang SP, Emura T (2017) A Bayesian approach with generalized ridge estimation for high-dimensional regression and testing, Commun Stat-Simul 46(8): 6083-105.

Hoerl AE, Kennard RW (1970) Ridge regression: Biased estimation for nonorthogonal problems. Technometrics 12:55–67.

Examples

n=100 # no. of observations
p=100 # no. of dimensions
q=r=10 # no. of nonzero coefficients
beta=c(rep(0.5,q),rep(0.5,r),rep(0,p-q-r))
X=X.mat(n,p,q,r)
Y=X%*%beta+rnorm(n,0,1)
g.ridge(X,Y-mean(Y),method="HK",kmax=200)
g.ridge(X,Y-mean(Y),method="YE",kmax=200)