Parameter Estimation Of Linear Model Under Decision Theoretic Analysis

The principle of least squares is very important.It reflects the goodness of fit. The least squares solution is easy to get and has a concise form. The known Gauss-Markov theorem asserts that the least squares estimator(LSE) is a minimum variance unbiased estimator among the class of all linear unbiased estimators. These properties give the LSE extensive use in parameter estimation theory and application. In the near 20 years, many statistician have made the extensions to the Gauss-Markov theorem from different viewpoints and studied the optimality of LSE under kinds of criteria. These extensions not only have vital significance in statistical theory and application ,but also full of mathematical beauty. On the other hand, statisticians study the improvements on the LSE under all kinds of models and propose many new estimators. This paper studies the performance and the optimality of estimators in decision theoretic analysis,including the robustness of LSE of linear model;the restricted LSE of the regression coefficients β with equality and inequality linear constraints on the parameter space;and improved confidence intervals for the error variance of the normal linear model .There are 4 chapters .The main results are summarized as follow:In chapter 1,a brief review of developments of the finite sample performance of LSE are presented.The evaluation of estimators under decision theoretic framework will always involve the distributions of the statistics,the choice of the class of estimation and loss functions,and the criteria for comparison of estimators.we suppose the distribution of the errors lying in the class of elliptically symmetric distributions here and there in this paper. So we briefly review the class of elliptically symmetric distributions and some of its properties.We also introduce a new and strong criterion- the concentration probability criterion which implies the domination under the mean squared error criterion. In this paper ,we will adopt concentration probability as a criterion for comparison of estimators.The last section introduces the asymmetric loss functions and balanced loss functions.Chapter 2 deals with the robustness of LSE from the respective of the distributions of the error terms and extends the Gauss-Markov theorem. Consider the linear regression modely=Xβ+ε...