Admissibility Of Parameter Estimation On Linear Model

This paper discusses the admissibility of parameter estimation based on linear model. On the linear model: Y = Xβ+ε, E (ε) = 0, Cov (ε)=σ2V(one variable model), as to the estimation of the parameter Sβ, people provides many results, in which one important problem is discussing their properties, and the admissibility is a very basic requirement to a certain content. To this problem, people divide it into two parts: one is that the admissibility of the linear estimation in the class of linear estimators. The other is the admissibility of the linear estimation in the class of all estimators. In the discussion, we can find that, to the relation of the two parts, normal distribution play an important role. Before the discussion, we can see the relationship of the three kinds of loss function: quadratic loss, matrix loss, vector loss. And we can find the differences of conditions in each admissibility range clearly. The sequence that this paper discusses the linear model is from special model to the general model. First, we discuss that the necessary and sufficient conditions for a estimator to be admissible in one-variable linear model based on three kinds of loss functions. I give the results on special model based on V = In, subsequently we focus on the general discussion. In the main body, we discuss the admissibility of homogeneous linear estimator LY , after that, we study the case of nonhomogeneous linear estimator LY +α. And then we discuss the corresponding problems in multi-linear model. In addition, when we solve actual problems, we would meet some constrained linear model. While we know the constrained linear model can transfer into the unconstrained linear model, so this paper focuses on the unconstrained linear model. This paper consists of five parts. At first we introduce the theory of admissibility, significance and its current situations. And then we provide pre-knowledge, including the model introduction, some concept and formulas. The part of following is the main body, and the sequence of discussing is from the one variable linear model to the multi-linear model and from the class of homogeneous to nonhomogeneous. At last we obtain the conclusion, and summarize current fruits and the innovative points.