On The Estimation Of Normal Linear Model Under Balanced Loss Functions

This article mainly discusses about parameter estimation in linear models. In the field of parameter estimation, we often use the statistics tool-loss function to pass judgments on the goodness of the parameter estimators. And when talk about linear model, the square loss i.e. Mean Square Error is the traditional method. This loss function reflects the precision of estimation portion. If we regard the Residual Sum of Squares which reflects the fitness of the linear model as a criterion to judge parameter estimators, naturally we hope the estimator could get good results under the two kinds of criterions. Just because of the reason, Zellner proposes the Balanced Loss Functions to integrate the goodness-of-fit portion and the precision of estimation portion.The key point of this article is to use the Balanced Loss Functions to discuss one of the familiar estimators-the Ridge Estimator, comparing to the Least Square Estimator, which is the most useful and basic estimator in linear model. The problem will be discussed by three steps. Firstly, we introduce the statistics model discussed, that is normal linear model, and the key concept- the Balanced Loss Functions. Secondly we give the estimators mentioned and correlated concepts. In virtue of the Least Square Estimator under the multicollinear of data, we introduce the Ridge Estimator, and generalize it. The last part mainly discusses the goodness of the Ridge Estimator and the relative estimators under Balanced Loss Functions, and gets some results about the admissibility and dominant of the estimators. To illuminate the estimators and the loss function, computer simulation is used.