The Superiority Of The Liu-type Principal Components Estimate

In the linear regression problem, the least squares estimate is the most widely used estimate. However, when multicollinearity between variables, the least squares estimate will be limited. For this, K.J.Liu brings a new unbiased estimate-Liu estimate. Liu estimate plays better than the least squares estimate when the design matrix means weak. This paper reaserch Liu-type principal components estimate’s superiority under the balance loss function, meanwhile, investigate some properties of the generalized Liu-type principal components estimate.This paper first outlines the basic research situation of parameter estimation. The second chapter describes some of the basics of this article will be involved in, such as the linear model, least squares estimate, admissibility and balancee loss function. Chapter III studys some properties of the Liu-type principal component estimate, and proves that Liu-type principal component estimate’s admissibility. Fourth chapter, we will see Liu-type principal component estimate’s superiority under the balance loss function, and prove that Liu-type principal component estimate is better than least squares estimate principal component estimate and Liu estimate under certain conditions, gives numerical simulation at the end.At the Chapter Five, we would see generalized Liu-type principal components estimate, prove that it is better than the least squares estimate in the mean square error sense under certain conditions and prove a generalized Liu-type principal components estimate’s superiority. Finally, the paper shows that on the generalized Liu-type principal component estimate’s superiority to the least squares estimate when the design matrix is weak by the actual data.