This thesis investigates improved estimators of the parameters of the linear regression models with normal errors, under sample and non-sample prior information about the value of the parameters. The estimators considered are the unrestricted estimator (UE), restricted estimator (RE), shrinkage restricted estimator (SRE), preliminary test estimator (PTE), shrinkage preliminary test estimator (SPTE), and shrinkage estimator (SE). The performances of the estimators are investigated with respect to bias, squared error and linex loss. For the analyses of the risk functions of the estimators, analytical, graphical and numerical procedures are adopted. In Part I the SRE, SPTE and SE of the slope and intercept parameters of the simple linear regression model are considered. The performances of the estimators are investigated with respect to their biases and mean square errors. The efficiencies of the SRE, SPTE and SE relative to the UE are obtained. It is revealed that under certain conditions, SE outperforms the other estimators considered in this thesis. In Part II in addition to the likelihood ratio (LR) test, the Wald (W) and Lagrange multiplier (LM) tests are used to define the SPTE and SE of the parameter vector of the multiple linear regression model with normal errors. Moreover, the modified and size-corrected W, LR and LM tests are used in the definition of SPTE. It is revealed that a great deal of conflict exists among the quadratic biases (QB) and quadratic risks (QR) of the SPTEs under the three original tests. The use of the modified tests reduces the conflict among the QRs, but not among the QBs. However, the use of the size-corrected tests in the definition of the SPTE almost eliminates the conflict among both QBs and QRs. It is also revealed that there is a great deal of conflict among the performances of the SEs when the three original tests are used as the preliminary test statistics. With respect to quadratic bias, the W test statistic based SE outperforms that based on the LR and LM test statistics. However, with respect to the QR criterion, the LM test statistic based SE outperforms the W and LM test statistics based SEs, under certain conditions. In Part III the performance of the PTE of the slope parameter of the simple linear regression model is investigated under the linex loss function. This is motivated by increasing criticism of the squared error loss function for its inappropriateness in many real life situations where underestimation of a parameter is more serious than its overestimation or vice-versa. It is revealed that under the linex loss function the PTE outperforms the UE if the nonsample prior information about the value of the parameter is not too far from its true value. Like the linex loss function, the risk function of the PTE is also asymmetric. However, if the magnitude of the scale parameter of the linex loss is very small, the risk of the PTE is nearly symmetric.

In this thesis different situations are considered in which the preliminary test estimator is applied and the performance of the preliminary test estimator under different proposed loss functions, namely the reflected normal , linear exponential (LINEX) and bounded LINEX (BLINEX) loss functions is evaluated. In order to motivate the use of the BLINEX loss function rather than the reflected normal loss or the LINEX loss function, the risk for the preliminary test estimator and its component estimators derived under BLINEX loss is compared to the risk of the preliminary test estimator and its components estimators derived under both reflected normal loss and LINEX loss analytically (in some sections) and computationally. It is shown that both the risk under reflected normal loss and the risk under LINEX loss is higher than the risk under BLINEX loss. The key focus point under consideration is the estimation of the regression coefficients of a multiple regression model under two conditions, namely the presence of multicollinearity and linear restrictions imposed on the regression coefficients. In order to address the multicollinearity problem, the regression coefficients were adjusted by making use of Hoerl and Kennard’s (1970) approach in ridge regression. Furthermore, in situations where under- or overestimation exist, symmetric loss functions will not give optimal results and it was necessary to consider asymmetric loss functions. In the economic application, it was shown that a loss function which is both asymmetric and bounded to ensure a maximum upper bound for the loss, is the most appropriate function to use. In order to evaluate the effect that different ridge parameters have on the estimation, the risk values were calculated for all three ridge regression estimators under different conditions, namely an increase in variance, an increase in the level of multicollinearity, an increase in the number of parameters to be estimated in the regression model and an increase in the sample size. These results were compared to each other and summarised for all the proposed estimators and proposed loss functions. The comparison of the three proposed ridge regression estimators under all the proposed loss functions was also summarised for an increase in the sample size and an increase in variance. / Thesis (PhD)--University of Pretoria, 2014. / lk2014 / Statistics / PhD / Unrestricted