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

Η παρούσα μεταπτυχιακή διατριβή εντάσσεται ερευνητικά στην περιοχή της Στατιστικής Θεωρίας Αποφάσεων και ειδικότερα στην εκτίμηση των παραμέτρων στο μοντέλο της διπαραμετρικής εκθετικής κατανομής με παράμετρο θέσης μ και παράμετρο κλίμακος σ. Θεωρούμε το πρόβλημα εκτίμησης των παραμέτρων κλίμακας μ και θέσης σ, όταν μ≤c, όπου c είναι μία γνωστή σταθερά. Αποδεικνύουμε ότι σε σχέση με το κριτήριο του Μέσου Τετραγωνικού Σφάλματος (ΜΤΣ), οι βέλτιστοι αναλλοίωτοι εκτιμητές των μ και σ, είναι μη αποδεκτοί όταν μ≤c, και προτείνουμε βελτιωμένους. Επίσης συγκρίνουμε του εκτιμητές αυτούς σε σχέση με το κριτήριο του Pitman. Επιπλέον, προτείνουμε εκτιμητές που είναι καλύτεροι από τους βέλτιστους αναλλοίωτους εκτιμητές, όταν μ≤c, ως προς την συνάρτηση ζημίας LINEX. Τέλος, η θεωρία που αναπτύσσεται εφαρμόζεται σε δύο ανεξάρτητα δείγματα προερχόμενα από εκθετική κατανομή. / The present master thesis deals with the estimation of the location parameter μ and the scale parameter σ of the two-parameter exponential distribution. We consider the problem of estimation of locasion parameter μ and the scale parameter σ, when it is known apriori that μ≤c, where c is a known constant. We establish that with respect to the mean square error (mse) criterion the best affine estimators of μ and σ in the absence of information μ≤c are inadmissible and we propose estimators which are better than these estimators. Also, we compare these estimators with respect to the Pitman Nearness criterion. We propose estimators which are better than the standard estimators in the unrestricted case with respect to the suitable choise of LINEX loss. Finally, the theory developed is applied to the problem of estimating the location and scale parameters of two exponential distributions when the location parameters are ordered.