# Test-Statistics This function calculates the test statistics, assuming $\mathcal{H}_0: H \beta = h$. When the error has a normal distribution, it is defined as [REMOVE_ME]$$ \mathcal{L} = \frac{(H\hat{\beta}^{U}-h)^{\top}(H(X^{\top}X)^{-1}H^{\top})^{-1}(H\hat{\beta}^{U}-h) }{q s^2_{unr}} [REMOVE_ME_2]$$ and when the error has a non-normal distribution, as [REMOVE_ME]$$ \mathcal{L} = \frac{(H\hat{\beta}^{U}-h)^{\top}(H(X^{\top}X)^{-1}H^{\top})^{-1}(H\hat{\beta}^{U}-h) }{s^2_{unr}} [REMOVE_ME_2]$$ 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 $\sigma^2$. ```r 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`. ## Returns The value of the test statistic. ## 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 $\sigma^2$. ## Examples ```r 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) ``` ## 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.

teststat function