# The Preliminary Test Estimator This function calculates the preliminary test. When the error has a normal distribution, the test statistic is given by [REMOVE_ME]$$ \hat{\beta}^{PT}=\hat{\beta}^{U} - (\hat{\beta}^{U} - \hat{\beta}^{R}) I(\mathcal{L} \le F_{q,n-p}(\alpha)) [REMOVE_ME_2]$$ and, if the error has a non-normal distribution, is given by [REMOVE_ME]$$ \hat{\beta}^{PT}=\hat{\beta}^{U} - (\hat{\beta}^{U} - \hat{\beta}^{R}) I(\mathcal{L} \le \chi^2_{q}(\alpha)) [REMOVE_ME_2]$$ 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. ```r 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`. ## Returns 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. ## 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. ## Details The corresponding estimator of $\sigma^2$ is $$ s^2 = \frac{1}{n-p}(y-X\hat{\beta}^{PT})^{\top}(y - X\hat{\beta}^{PT}). $$ ## 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)) 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) ``` ## 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.

ptReg function