Package 'ImpShrinkage'

Hald's Cement Data

Description

Heat evolved (cals/gm) in the setting of 13 samples of Portland cement with different percentage weight of chemical components.

Format

A data.frame with 13 observations on the following 5 variables.

x1

percentage weight in clinkers of 3CaO.Al2O3

x2

percentage weight in clinkers of 3CaO.SiO2

x3

percentage weight in clinkers of 4CaO.Al2O3.Fe2O3

x4

percentage weight in clinkers of 2CaO.SiO2

y

heat evolved (calories/gram)

Source

Woods, H., Steinour, H. H. and Starke, H. R. (1932) Effect of composition of Portland cement on heat evolved during hardening. Industrial Engineering and Chemistry, 24, 1207–1214.

Examples

data("cement")
cement

---

Extract Model Coefficients

Description

Coefficients extracted from the model object improvedpreliminaryTest

Usage

## S3 method for class 'improvedpreliminaryTest'
coefficients(object, ...)

## S3 method for class 'improvedpreliminaryTest'
coef(object, ...)

Arguments

object	An object of class improvedpreliminaryTest.
...	Other arguments.

Value

A vector of coefficients.

See Also

coefficients.unrestricted, coefficients.restricted, coefficients.preliminaryTest, coefficients.stein, coefficients.positivestein, coef.unrestricted, coef.restricted, coef.positivestein, coef.stein, coef.positivestein.

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))
model <- iptReg(X, y, H, h, alpha = 0.05)
coefficients(model)
coef(model)

---

Extract Model Coefficients

Description

Coefficients extracted from the model object positivestein

Usage

## S3 method for class 'positivestein'
coefficients(object, ...)

## S3 method for class 'positivestein'
coef(object, ...)

Arguments

object	An object of class positivestein.
...	Other arguments.

Value

A vector of coefficients.

See Also

coefficients.unrestricted, coefficients.restricted, coefficients.preliminaryTest, coefficients.improvedpreliminaryTest, coefficients.stein, coef.unrestricted, coef.restricted, coef.preliminaryTest, coef.improvedpreliminaryTest, coef.stein.

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))
model <- prstReg(X, y, H, h)
coefficients(model)
coef(model)

---

Extract Model Coefficients

Description

Coefficients extracted from the model object preliminaryTest

Usage

## S3 method for class 'preliminaryTest'
coefficients(object, ...)

## S3 method for class 'preliminaryTest'
coef(object, ...)

Arguments

object	An object of class preliminaryTest.
...	Other arguments.

Value

A vector of coefficients.

See Also

coefficients.unrestricted, coefficients.restricted, coefficients.improvedpreliminaryTest, coefficients.stein, coefficients.positivestein, coef.unrestricted, coef.restricted, coef.improvedpreliminaryTest. coef.stein, coef.positivestein. #'

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))
model <- ptReg(X, y, H, h, alpha = 0.05)
coefficients(model)
coef(model)

---

Extract Model Coefficients

Description

Coefficients extracted from the model object restrcited.

Usage

## S3 method for class 'restricted'
coefficients(object, ...)

## S3 method for class 'restricted'
coef(object, ...)

Arguments

object	An object of class restricted.
...	Other arguments.

Value

A vector of coefficients.

See Also

coefficients.unrestricted, coefficients.preliminaryTest, coefficients.improvedpreliminaryTest, coefficients.stein, coefficients.positivestein, coef.unrestricted, coef.preliminaryTest, coef.improvedpreliminaryTest coef.stein, coef.positivestein.

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))
model <- resReg(X, y, H, h)
coefficients(model)
coef(model)

---

Extract Model Coefficients

Description

Coefficients extracted from the model object stein

Usage

## S3 method for class 'stein'
coefficients(object, ...)

## S3 method for class 'stein'
coef(object, ...)

Arguments

object	An object of class stein.
...	Other arguments.

Value

A vector of coefficients.

See Also

coefficients.unrestricted, coefficients.restricted, coefficients.preliminaryTest, coefficients.improvedpreliminaryTest, coefficients.positivestein, coef.unrestricted, coef.restricted, coef.preliminaryTest, coef.improvedpreliminaryTest, coef.positivestein.

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), nr = 3, nc = p, byrow = TRUE)
h <- rep(0, nrow(H))
model <- stReg(X, y, H, h)
coefficients(model)
coef(model)

---

Extract Model Coefficients

Description

Coefficients extracted from the model object unrestricted.

Usage

## S3 method for class 'unrestricted'
coefficients(object, ...)

## S3 method for class 'unrestricted'
coef(object, ...)

Arguments

object	An object of class unrestricted.
...	Other arguments.

Value

A vector of coefficients.

See Also

coefficients.restricted, coefficients.preliminaryTest, coefficients.improvedpreliminaryTest, coefficients.stein, coefficients.positivestein, coef.restricted, coef.preliminaryTest, coef.improvedpreliminaryTest coef.stein, coef.positivestein.

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
model <- unrReg(X, y)
coefficients(model)
coef(model)

---

Extract Model Fitted Values

Description

Fitted values based on object improvedpreliminaryTest.

Usage

## S3 method for class 'improvedpreliminaryTest'
fitted(object, ...)

Arguments

object	An object of class improvedpreliminaryTest.
...	Other arguments.

Value

A vector of fitted values.

See Also

fitted.unrestricted, fitted.restricted, fitted.preliminaryTest, fitted.stein, fitted.positivestein.

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))
model <- iptReg(X, y, H, h, alpha = 0.05)
fitted(model)

---

Extract Model Fitted Values

Description

Fitted values based on object positivestein.

Usage

## S3 method for class 'positivestein'
fitted(object, ...)

Arguments

object	An object of class positivestein.
...	Other arguments.

Value

A vector of fitted values.

See Also

fitted.unrestricted, fitted.restricted, fitted.preliminaryTest, fitted.improvedpreliminaryTest, fitted.stein.

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))
model <- prstReg(X, y, H, h)
fitted(model)

---

Extract Model Fitted Values

Description

Fitted values based on object preliminaryTest.

Usage

## S3 method for class 'preliminaryTest'
fitted(object, ...)

Arguments

object	An object of class preliminaryTest.
...	Other arguments.

Value

A vector of fitted values.

See Also

fitted.unrestricted, fitted.restricted, fitted.improvedpreliminaryTest, fitted.stein, fitted.positivestein

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))
model <- ptReg(X, y, H, h, alpha = 0.05)
fitted(model)

---

Extract Model Fitted Values

Description

Fitted values based on object restrcited.

Usage

## S3 method for class 'restricted'
fitted(object, ...)

Arguments

object	An object of class restricted.
...	Other arguments.

Value

Fitted values extracted from the object restricted.

See Also

fitted.unrestricted, fitted.preliminaryTest, fitted.improvedpreliminaryTest, fitted.stein, fitted.positivestein

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))
model <- resReg(X, y, H, h)
fitted(model)

---

Extract Model Fitted Values

Description

Fitted values based on object stein.

Usage

## S3 method for class 'stein'
fitted(object, ...)

Arguments

object	An object of class stein.
...	Other arguments.

Value

A vector of fitted values.

See Also

fitted.unrestricted, fitted.restricted, fitted.preliminaryTest, fitted.improvedpreliminaryTest, fitted.positivestein

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), nr = 3, nc = p, byrow = TRUE)
h <- rep(0, nrow(H))
model <- stReg(X, y, H, h)
fitted(model)

---

Extract Model Fitted Values

Description

Fitted values based on object unrestricted.

Usage

## S3 method for class 'unrestricted'
fitted(object, ...)

Arguments

object	An object of class unrestricted.
...	Other arguments.

Value

A vector of fitted values.

See Also

fitted.restricted, fitted.preliminaryTest, fitted.improvedpreliminaryTest, fitted.stein, fitted.positivestein.

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
model <- unrReg(X, y)
fitted(model)

---

The improved preliminary test estimator

Description

This function calculates the improved preliminary test estimator. When the error has a normal distribution, this estimator can be calculated by

$\hat{\beta}^{iPT}= \hat{\beta}^{PT} - d (\hat{\beta}^{U} - \hat{\beta}^{R}) \mathcal{L}^{-1} I(\mathcal{L} > F_{q,n-p}(\alpha))$

and, when the error has a non-normal distribution, by

$\hat{\beta}^{iPT}= \hat{\beta}^{PT} - d (\hat{\beta}^{U} - \hat{\beta}^{R}) \mathcal{L}^{-1} I(\mathcal{L} > \chi^2_{q}(\alpha))$

where $I(A)$ denotes an indicator function and

• $\hat{\beta}^{PT}$ is the preliminary test estimator; See ptReg

• $\hat{\beta}^{U}$ 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;

• $d$ is the shrinkage factor;

• $\alpha$ is the significance level.

Usage

iptReg(X, y, H, h, alpha, 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.
alpha	A given significance level.
d	(optional) If not provided (or set to NULL), it will be calculated using $\frac{{(q - 2) \cdot (n - p)}}{{q \cdot (n - p + 2)}}.$
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 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

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

Value

An object of class improvedpreliminaryTest 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))
iptReg(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))
iptReg(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))
iptReg(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))
iptReg(X, y, H, h, alpha = 0.05)

---

Extract Model Predictions Values

Description

Predicted values based on object improvedpreliminaryTest.

Usage

## S3 method for class 'improvedpreliminaryTest'
predict(object, newdata, ...)

Arguments

object	An object of class "improvedpreliminaryTest".
newdata	An optional data frame in which to look for variables with which to predict. If omitted, the fitted values are used.
...	Other arguments.

Value

A vector of predictions.

See Also

predict.unrestricted, predict.restricted, predict.preliminaryTest, predict.stein, predict.positivestein.

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))
model <- iptReg(X, y, H, h, alpha = 0.05)
predict(model, X)

---

Extract Model Predictions Values

Description

Predicted values based on object positivestein.

Usage

## S3 method for class 'positivestein'
predict(object, newdata, ...)

Arguments

object	An object of class "positivestein".
newdata	An optional data frame in which to look for variables with which to predict. If omitted, the fitted values are used.
...	Other arguments.

Value

A vector of predictions.

See Also

predict.unrestricted, predict.restricted, predict.preliminaryTest, predict.improvedpreliminaryTest, predict.stein.

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))
model <- prstReg(X, y, H, h)
predict(model, X)

---

Extract Model Predictions Values

Description

Predicted values based on object preliminaryTest.

Usage

## S3 method for class 'preliminaryTest'
predict(object, newdata, ...)

Arguments

object	An object of class "preliminaryTest".
newdata	An optional data frame in which to look for variables with which to predict. If omitted, the fitted values are used.
...	Other arguments.

Value

A vector of predictions.

See Also

predict.unrestricted, predict.restricted, predict.improvedpreliminaryTest, predict.stein, predict.positivestein.

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))
model <- ptReg(X, y, H, h, alpha = 0.05)
predict(model, X)

---

Extract Model Predictions Values

Description

Predicted values based on object restrcited.

Usage

## S3 method for class 'restricted'
predict(object, newdata, ...)

Arguments

object	An object of class restricted.
newdata	An optional data frame in which to look for variables with which to predict. If omitted, the fitted values are used.
...	Other arguments.

Value

A vector of predictions.

See Also

predict.unrestricted, predict.preliminaryTest, predict.improvedpreliminaryTest, predict.stein, predict.positivestein.

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))
model <- resReg(X, y, H, h)
predict(model, X)

---

Extract Model Predictions Values

Description

Predicted values based on object stein.

Usage

## S3 method for class 'stein'
predict(object, newdata, ...)

Arguments

object	An object of class "stein".
newdata	An optional data frame in which to look for variables with which to predict. If omitted, the fitted values are used.
...	Other arguments.

Value

A vector of predictions.

See Also

predict.unrestricted, predict.restricted, predict.preliminaryTest, predict.improvedpreliminaryTest, predict.positivestein.

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), nr = 3, nc = p, byrow = TRUE)
h <- rep(0, nrow(H))
model <- stReg(X, y, H, h)
predict(model, X)

---

Extract Model Predictions Values

Description

Predicted values based on object unrestricted.

Usage

## S3 method for class 'unrestricted'
predict(object, newdata, ...)

Arguments

object	An object of class unrestricted.
newdata	An optional data frame in which to look for variables with which to predict. If omitted, the fitted values are used.
...	Other arguments.

Value

A vector of predictions.

See Also

predict.restricted, predict.preliminaryTest, predict.improvedpreliminaryTest, predict.stein, predict.positivestein.

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
model <- unrReg(X, y)
predict(model, X)

---

The positive-rule Stein estimator

Description

This function calculates the positive-rule Stein estimator. This estimator is an improved version of the Stein estimator, where only the positive part of the shrinking factor is considered. It may be calculated by

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

where $I(A)$ denotes an indicator function and

• $\hat{\beta}^{S}$ is the Stein estimator; See stReg.

• $\hat{\beta}^{U}$ 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

prstReg(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 calculated using $\frac{{(q - 2) \cdot (n - p)}}{{q \cdot (n - p + 2)}}.$
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 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 given by

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

Value

An object of class pst 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), nr = 3, nc = p, byrow = TRUE)
h <- rep(0, nrow(H))
prstReg(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), nr = 3, nc = p, byrow = TRUE)
h <- rep(1, nrow(H))
prstReg(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))
prstReg(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))
prstReg(X, y, H, h)

---

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)

---

Extract Model Residuals

Description

Residuals values based on model object improvedpreliminaryTest.

Usage

## S3 method for class 'improvedpreliminaryTest'
residuals(object, ...)

Arguments

object	An object of class improvedpreliminaryTest.
...	Other arguments.

Value

A vector of residuals.

See Also

residuals.unrestricted, residuals.restricted, residuals.preliminaryTest, residuals.stein, residuals.positivestein,

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))
model <- iptReg(X, y, H, h, alpha = 0.05)
residuals(model)

---

Extract Model Residuals

Description

Residuals values based on model object positivestein.

Usage

## S3 method for class 'positivestein'
residuals(object, ...)

Arguments

object	An object of class positivestein.
...	Other arguments.

Value

A vector of residuals.

See Also

residuals.unrestricted, residuals.restricted, residuals.preliminaryTest, residuals.improvedpreliminaryTest, residuals.stein.

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))
model <- prstReg(X, y, H, h)
residuals(model)

---

Extract Model Residuals

Description

Residuals values based on model object preliminaryTest.

Usage

## S3 method for class 'preliminaryTest'
residuals(object, ...)

Arguments

object	An object of class preliminaryTest.
...	Other arguments.

Value

A vector of residuals.

See Also

residuals.unrestricted, residuals.restricted, residuals.improvedpreliminaryTest, residuals.stein, residuals.positivestein.

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))
model <- ptReg(X, y, H, h, alpha = 0.05)
residuals(model)

---

Extract Model Residuals

Description

Residuals values based on model object restricted.

Usage

## S3 method for class 'restricted'
residuals(object, ...)

Arguments

object	An object of class restricted.
...	Other arguments.

Value

A vector of residuals.

residuals.unrestricted, residuals.preliminaryTest, residuals.improvedpreliminaryTest, residuals.stein, residuals.positivestein.

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))
model <- resReg(X, y, H, h)
residuals(model)

---

Extract Model Residuals

Description

Residuals values based on model object stein.

Usage

## S3 method for class 'stein'
residuals(object, ...)

Arguments

object	An object of class stein.
...	Other arguments.

Value

A vector of residuals.

See Also

residuals.unrestricted, residuals.restricted, residuals.preliminaryTest, residuals.improvedpreliminaryTest, residuals.positivestein,.

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), nr = 3, nc = p, byrow = TRUE)
h <- rep(0, nrow(H))
model <- stReg(X, y, H, h)
residuals(model)

---

Extract Model Residuals

Description

Residuals values based on model object unrestricted.

Usage

## S3 method for class 'unrestricted'
residuals(object, ...)

Arguments

object	An object of class unrestricted.
...	Other arguments.

Value

A vector of residuals.

See Also

residuals.restricted, residuals.preliminaryTest, residuals.improvedpreliminaryTest residuals.stein, residuals.positivestein.

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
model <- unrReg(X, y)
residuals(model)

---

The restricted estimator

Description

This function calculates the restricted estimator using

$\hat{\beta}^{R} = \hat{\beta}^{U} - (X^{\top}X)^{-1}H^{\top} (H(X^{\top}X)^{-1}H^{\top})^{-1}(H\hat{\beta}^{U}-h)$

where

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

• $H\beta = h$ represents a subspace of the parameter space induced by the non-sample information. Here, $H$ is a known $q \times p$ matrix, and $h$ is a known $q$-vector.

Usage

resReg(X, y, H, h)

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.

Details

#' The corresponding estimator of $\sigma^2$ is

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

Value

An object of class restricted 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))
resReg(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))
resReg(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))
resReg(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))
resReg(X, y, H, h)

---

Simulation data

Description

This function generates a toy example. The error term, $\varepsilon$, and the design matrix, $X$, are simulated from standard normal distributions, $\mathcal{N}(0,1)$, using the rnorm function. Given the true parameter vector, $\beta$, the response vector, $y$, is calculated as

$y = X \beta + \varepsilon.$

Usage

simdata(n, p, beta, seed = NULL)

Arguments

n	Number of observations.
p	Number of variables.
beta	Regression parameter.
seed	(Optional) The random seed for reproducibility. Default is NULL.

Value

A list containing the following components:

X

a matrix of dimensions n x p.

y

a numeric vector of length n.

References

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

Examples

simulated_data <- simdata(n = 100, p = 5, beta = c(2, 1, 3, 0, 5))
X <- simulated_data$X
y <- simulated_data$y
X
y

---

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)

---

Test-Statistics

Description

This function calculates the test statistics, assuming $\mathcal{H}_0: H \beta = h$. When the error has a normal distribution, it is defined as

$\mathcal{L} = \frac{(H\hat{\beta}^{U}-h)^{\top}(H(X^{\top}X)^{-1} H^{\top})^{-1}(H\hat{\beta}^{U}-h) }{q s^2_{unr}}$

and when the error has a non-normal distribution, as

$\mathcal{L} = \frac{(H\hat{\beta}^{U}-h)^{\top}(H(X^{\top}X)^{-1} H^{\top})^{-1}(H\hat{\beta}^{U}-h) }{s^2_{unr}}$

where

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

• $q$ is the number of restrictions, i.e., the number of rows of known matrix $H$;

• $s^2_{unr}$ is the corresponding unrestricted estimator of $\sigma^2$.

Usage

teststat(X, y, H, h, 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.
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 the F distribution (FDist).On the other hand, if the errors have a non-normal distribution, the asymptotic distribution of the test statistics is the $\chi^2$ distribution (Chisquare). By default, is_error_normal is set to FALSE.

Value

The value of the test statistic.

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_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))
teststat(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))
teststat(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))
teststat(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))
teststat(X, y, H, h)

---

The Unrestricted estimator

Description

This function calculates the unrestricted estimator as

$\hat{\beta}^{U} = (X^{\top} X)^{-1} X^{\top} y$

where $^{\top}$ denotes the transpose of a matrix. It is important to note that the input matrices $X$ and $y$ should be standardized, for example, by using scale. Alternatively, the user can employ lm to obtain this estimator, but it is crucial to remember to set intercept = FALSE.

Usage

unrReg(X, y)

Arguments

X	Matrix with input observations, of dimension n x p, where each row is an observation vector;
y	Vector with response observations of size n.

Details

The corresponding unrestricted estimator of $\sigma^2$ is

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

Value

An object of class unrestricted 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.

Examples

data(cement)
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
unrReg(X, y)

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))
unrReg(X, y)

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))
unrReg(X, y)

---