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.

The present thesis provides a general asymptotic theory for preliminary test estimators (PTEs). PTEs are typically used when one needs to estimate a parameter having some uncertain prior information about it. In the literature, preliminary test estimation have been applied to some specific models, but no general asymptotic theory was already available to the best of our knowledge. After a study of PTEs in a multisample principal component context, we first provide a general asymptotic theory for PTEs in uniformly, locally and asymptotically normal (ULAN) models. An extensive list of statistical and econometric models are ULAN making our results quite general. Our main results are obtained using the Le Cam asymptotic theory under the assumption that the estimators involved in the PTEs admit Bahadur-type asymptotic representations. Then, we propose PTEs involving multiple tests and therefore multiple constrained estimators; we call them preliminary multiple test estimators. For the latter, we also derive a very general asymptotic theory in ULAN models. Our theoretical results are illustrated on problems involving the estimation of covariance matrices both via simulations and a real data example. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

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

<p>This thesis investigates the statistical properties of preliminary test estimators of linear models with normally distributed errors. Specifically, we derive exact expressions for the mean, variance and quadratic risk (i.e. the Mean Square Error) of estimators whose form are determined by the outcome of a statistical test. In the process, some new results on the moments of truncated linear or quadratic forms in normal vectors are established.</p><p>In the first paper (Paper I), we consider the estimation of the vector of regression coefficients under a model selection procedure where it is assumed that the analyst chooses between two nested linear models by some of the standard model selection criteria. This is shown to be equivalent to estimation under a preliminary test of some linear restrictions on the vector of regression coefficients. The main contribution of Paper I compared to earlier research is the generality of the form of the test statistic; we only assume it to be a quadratic form in the (translated) observation vector. Paper II paper deals with the estimation of the regression coefficients under a preliminary test for homoscedasticity of the error variances. In Paper III, we investigate the statistical properties of estimators, truncated at zero, of variance components in linear models with random effects. Paper IV establishes some new results on the moments of truncated linear and/or quadratic forms in normally distributed vectors. These results are used in Papers I-III. In Paper V we study some algebraic properties of matrices that occur in the comparison of two nested models. Specifically we derive an expression for the inertia (the number of positive, negative and zero eigenvalues) of this type of matrices.</p>

This thesis investigates the statistical properties of preliminary test estimators of linear models with normally distributed errors. Specifically, we derive exact expressions for the mean, variance and quadratic risk (i.e. the Mean Square Error) of estimators whose form are determined by the outcome of a statistical test. In the process, some new results on the moments of truncated linear or quadratic forms in normal vectors are established. In the first paper (Paper I), we consider the estimation of the vector of regression coefficients under a model selection procedure where it is assumed that the analyst chooses between two nested linear models by some of the standard model selection criteria. This is shown to be equivalent to estimation under a preliminary test of some linear restrictions on the vector of regression coefficients. The main contribution of Paper I compared to earlier research is the generality of the form of the test statistic; we only assume it to be a quadratic form in the (translated) observation vector. Paper II paper deals with the estimation of the regression coefficients under a preliminary test for homoscedasticity of the error variances. In Paper III, we investigate the statistical properties of estimators, truncated at zero, of variance components in linear models with random effects. Paper IV establishes some new results on the moments of truncated linear and/or quadratic forms in normally distributed vectors. These results are used in Papers I-III. In Paper V we study some algebraic properties of matrices that occur in the comparison of two nested models. Specifically we derive an expression for the inertia (the number of positive, negative and zero eigenvalues) of this type of matrices.

The purpose of this paper is to investigate the topic of school-matureness diagnostics.Through theoretical background it presents, at which level a child should be developed enough in terms of his/her character and knowledge, before entering the compulsory school attendance. Terms such as school knowledge and school readiness are tightly connected to the topic and explained in this research. Recent and former legislative requirements concerning the beginning of one's compulsory school attendance, school enrollment, and postponement of compulsory school attendance are mentioned. Following the theoretical part, the practical part of this thesis diagnoses school maturity. The Preliminary test of school maturity by Jaroslav Jirasek is described in detail, other often used diagnostically materials and publicly available publications with content being well usable for the diagnoses are also shortly introduced. Base don those publications a test for the diagnosis of school maturity was developed, and it was together with Jirasek's Preliminary test of school maturity used for testing school maturity with a research sample. The results were analyzed and compared. The research focus was oriented on the applicability of Preliminary test of school maturity by Jaroslav Jirasek. Its results were compared with...

The diploma thesis deals with the issue of school maturity and readiness of pupils in the preparatory class. In the theoretical part the basic terminology from the field of pre-school education and problems of entering the elementary school is defined. There are also the principles of the preliminary test of school maturity described in detail. They are subsequently applied in practical research. The research part focuses on the comparison of the level of school maturity and readiness of pupils in the preparatory class and children in kindergarten. The survey was mainly conducted using qualitative methods, specifically through interviews, questionnaires and anamnesis. However, one quantitative method was also included - the standardized preliminary test of school maturity. This test focuses on fine motor skills and visuo-motor coordination ability. The research was carried out in preparatory classes and kindergartens in Mladá Boleslav and its surroundings. The goal of this diploma thesis was to find out whether education in a certain kind of preschool facility can affect children's readiness for school. Research results showed, that preparatory classes have better conditions for the development of a preschool pupil with postponement of school attendance than kindergartens. KEY WORDS Preschool...