Theory On The Admissible Estimators In Linear Models With Respect To Inequality Constraints

Linear model is a very important class of statistical models, which includes the linear regression model, analysis of variance model and so on. The linear models studied in this paper are only the linear regression models. How to estimate the unknown parameters and evaluate the quality of it, which is the parameter estimation theory, is a major issue to be resolved in linear model, and the admissiblility problem is the basic requirement to an estimator, therefore, it occupies important position in the parameter estimation theory. Currently, there have been quite fruitful results on the admissibility of the regression parameter in the linear regression model, but few results in linear regression model with respect to the inequality constraints. In this paper, we systematic study the admissiblility problem for the linear estimator of the regression parameters in linear regression model with respect to the inequality constraints.The thesis is divided into six chapters.The first chapter is the introduction part, which mainly introduces the linear model, the background and status of related research on the admissiblility of the parameter estimator in linear model, and the main results are also listed.In Chapter II, we study the admissibility for linear estimator of regression parameters in the simple linear model with respect to inequality constraints, under the balance loss function. In the class of homogeneous linear estimators, we prove that a linear estimator of the estimated function is admissible estimator in the simple linear model with respect to the inequality constraints is equivalent to that in the simple linear model, and give the necessary and sufficient conditions for linear estimators being admissible. In the class of non-homogeneous linear estimators, we find the relationship of the admissiblility of linear estimator between this two classes, and its relationship with the conditions of inequality constraints, the necessary and sufficient conditions for linear estimators being admissible in this class are also obtained.In Chapter III, we study the admissibility for linear estimator of regression parameters in the multivariate linear model with respect to inequality constraints, under the quadratic matrix loss function and the balance loss function respectively. In the class of homogeneous linear estimators, we give the necessary and sufficient condition for linear estimators being admissible. In the class of non-homogeneous linear estimators, when the model without constraints, under the quadratic matrix loss and quadratic loss function, the relevant research shows that:the admissibility of parameter estimators in these two losses is equivalent. However, when the model with respect to some inequality constraints, this equivalence no longer valid. We also give the necessary and sufficient conditions for linear estimators being admissible in the homogeneous and non-homogeneous classes respectively, under the balance loss function.In Chapter IV, we study the admissibility for linear estimator of regression parameters in the multivariate linear model with respect to inequality constraints under the matrix loss function. In the class of homogeneous linear estimators, in terms of an equivalence theorem, we give two forms of necessary and sufficient conditions for linear estimators of estimable function being admissible. Furthermore, we also studied the liner estimator of inestimable function, and obtain its necessary and sufficient condition. In the class of non-homogeneous linear estimators, we study the relationship of admissibility between the class of non-homogeneous and homogeneous linear estimators, find and correct some error results that exist in the former research paper, and obtain some results for linear estimators both of estimable and inestimable function being admissible.In Chapter V, by the theory of inequality and the vectorization transformation of a matrix, we study the admissibility problem of linear estimators in the growth curve model with respect to inequality constraints under the quadratic loss and the vector loss function. When the estimated function is estimable function, we give the necessary and sufficient conditions for linear estimators being admissible in the homogeneous and inhomogeneous classes respectively. We also studied the estimated function is inestimable function, and obtain the necessary and sufficient conditions for linear estimators of inestimable function being admissible.In the last chapter, we summarize the work of this paper, and present some related issues which is worth of further attention and study.