# The restricted estimator This function calculates the restricted estimator using [REMOVE_ME]$$ \hat{\beta}^{R} = \hat{\beta}^{U} - (X^{\top}X)^{-1}H^{\top}(H(X^{\top}X)^{-1}H^{\top})^{-1}(H\hat{\beta}^{U}-h) [REMOVE_ME_2]$$ 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. ```r 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. ## Returns 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. ## 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. ## Details #' The corresponding estimator of $\sigma^2$ is $$ s^2 = \frac{1}{n-p}(y-X\hat{\beta}^{R})^{\top}(y - X\hat{\beta}^{R}). $$ ## Examples ```r 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) ``` ## 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.

resReg function