ptReg {ImpShrinkage} | R Documentation |
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.
ptReg(X, y, H, h, alpha, is_error_normal = FALSE)
X |
Matrix with input observations, of dimension |
y |
Vector with response observations of size |
H |
A given |
h |
A given |
alpha |
A given significance level. |
is_error_normal |
logical value indicating whether the errors follow
a normal distribution. If |
The corresponding estimator of \sigma^2
is
s^2 = \frac{1}{n-p}(y-X\hat{\beta}^{PT})^{\top}(y - X\hat{\beta}^{PT}).
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.
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.
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)