ptReg {ImpShrinkage}

The Preliminary Test Estimator

Description

This function calculates the preliminary test. When the error has a normal distribution, the test statistic is given by

\hat{\beta}^{PT}=\hat{\beta}^{U} - (\hat{\beta}^{U} - \hat{\beta}^{R}) I(\mathcal{L} \le F_{q,n-p}(\alpha))

and, if the error has a non-normal distribution, is given by

\hat{\beta}^{PT}=\hat{\beta}^{U} - (\hat{\beta}^{U} - \hat{\beta}^{R}) I(\mathcal{L} \le \chi^2_{q}(\alpha))

where I(A) denotes an indicator function and

• \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;

• F_{q,n-p}(\alpha) is the upper \alpha level critical value of F-distribution with (q,n-p) degrees of freedom, calculated using qf;

• \chi^2_{q}(\alpha)is the upper \alpha level critical value of \chi^2-distribution with q degree of freedom, calculated using qchisq;

• \alpha: the significance level.

Usage

ptReg(X, y, H, h, alpha, 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.

alpha

A given significance level.

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 F distribution (FDist). On the other hand, if the errors have a non-normal distribution, the asymptotic distribution of the test statistics is \chi^2 distribution (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}^{PT})^{\top}(y - X\hat{\beta}^{PT}).

Value

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

coef

A named vector of coefficients.

residuals

The residuals, that is, the response values minus 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))
ptReg(X, y, H, h, alpha = 0.05)

# H beta != h
p <- ncol(X)
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))
ptReg(X, y, H, h, alpha = 0.05)

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))
ptReg(X, y, H, h, alpha = 0.05)
# 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))
ptReg(X, y, H, h, alpha = 0.05)