The Estimation Theory Of Regression Coefficient In Linear Regression Model Under Balanced Loss Function

On the basis of the balanced loss function proposed by Zellner, we discuss the balanced loss risk and obtain various estimations of regression coefficient in linear regression model.There are five chapters in this thesis. In Chapter 1, we introduce some theories and methods about the estimation of regression coefficient and do some preparations including some theories in the mean vector, the covariance matrix, the quadratic form of random vector, matrix derivative.In Chapter 2, based on the balanced loss risk, we propose the smallest risk estimation. We also get a nonlinear estimation (β|^)_w and discuss its statistical properties. Especially, we conclude that the nonlinear estimation (β|^)_w is better than the least squares estimationβ|^ in Pitman Closeness.In Chapter 3, we gain the admissible estimations of regression coefficient in several models including linear regression model, singular model and restricted regression model under the balanced loss function. At the same time, we obtain the Minimax estimation of regression coefficient in linear regression model and prove that it is unique.In Chapter 4, in the linear unbiased estimation category, we get the best linear unbiased estimation of the regression coefficient in linear regression model and singular model under the balanced loss function. When the covariance matrix is unknown, we give the definition of the relative efficiency of the best linear unbiased estimation based on the different standards, and obtain its upper (lower) bound.In Chapter 5, we obtain the Bayes estimations of regression coefficient under the balanced loss posterior risk by the three kinds of experiential distributions about the parameters.