stReg {ImpShrinkage}

The Stein estimator

Description

This function can be used to calculate the Stein estimator using

\hat{\beta}^{S}=\hat{\beta}^{U} - d \mathcal{L}^{-1} (\hat{\beta}^{U} - \hat{\beta}^{R})

where

• \hat{\beta}^{U} is the unrestricted estimator; See unrReg.

• \hat{\beta}^{R} is the restricted estimator; See resReg.

• \mathcal{L} is the test statistic. See teststat;

• d is the shrinkage factor.

Usage

stReg(X, y, H, h, d = NULL, is_error_normal = FALSE)

Arguments

X

Matrix with input observations, of dimension n x p; each row is an observation vector.

y

Vector with response observations of size n.

H

A given q x p matrix.

h

A given q x 1 vector.

d

(Optional) If not provided (or set to NULL), it will be set to be equal to \frac{{(q - 2) \cdot (n - p)}}{{q \cdot (n - p + 2)}}.

is_error_normal

logical value indicating whether the errors follow a normal distribution. If is_error_normal is TRUE, the distribution of the test statistics for the null hypothesis is FDist. On the other hand, if the errors have a non-normal distribution, the asymptotic distribution of the test statistics is Chisquare. By default, is_error_normal is set to FALSE.

Details

The corresponding estimator of \sigma^2 is

s^2 = \frac{1}{n-p}(y-X\hat{\beta}^{S})^{\top}(y - X\hat{\beta}^{S}).

Value

An object of class stein is a list containing at least the following components:

coef

A vector of coefficients.

residuals

The residuals, that is, the response values minus the fitted values.

s2

The estimated variance.

fitted.values

The fitted values.

References

Saleh, A. K. Md. Ehsanes. (2006). Theory of Preliminary Test and Stein‐Type Estimation With Applications, Wiley.

Kaciranlar, S., Akdeniz, S. S. F., Styan, G. P. & Werner, H. J. (1999). A new biased estimators in linear regression and detailed analysis of the widely-analysed dataset on portland cement. Sankhya, Series B, 61(3), 443-459.

Kibria, B. M. Golam (2005). Applications of Some Improved Estimators in Linear Regression, Journal of Modern Applied Statistical Methods, 5(2), 367- 380.

Examples

n_obs <- 100
p_vars <- 5
beta <- c(2, 1, 3, 0, 5)
simulated_data <- simdata(n = n_obs, p = p_vars, beta)
X <- simulated_data$X
y <- simulated_data$y
p <- ncol(X)
# H beta = h
H <- matrix(c(1, 1, -1, 0, 0, 1, 0, 1, 0, -1, 0, 0, 0, 1, 0), nrow = 3, ncol = p, byrow = TRUE)
h <- rep(0, nrow(H))
stReg(X, y, H, h)

# H beta != h
H <- matrix(c(1, 1, -1, 0, 0, 1, 0, 1, 0, -1, 0, 0, 0, 1, 0), nrow = 3, ncol = p, byrow = TRUE)
h <- rep(1, nrow(H))
stReg(X, y, H, h)

data(cement)
X <- as.matrix(cbind(1, cement[, 1:4]))
y <- cement$y
# Based on Kaciranlar et al. (1999)
H <- matrix(c(0, 1, -1, 1, 0), nrow = 1, ncol = 5, byrow = TRUE)
h <- rep(0, nrow(H))
stReg(X, y, H, h)
# Based on Kibria (2005)
H <- matrix(c(0, 1, -1, 1, 0, 0, 0, 1, -1, -1, 0, 1, -1, 0, -1), nrow = 3, ncol = 5, byrow = TRUE)
h <- rep(0, nrow(H))
stReg(X, y, H, h)