LIU-type Estimate In The Linear Model

The least square estimate is widely used in linear regression. However, the least square estimate does not always work well in some massive regressions, for there may be proximate linear correlation among the independent variables. To solve this problem, statisticians have suggested using biased estimate in place of unbiased estimate. Ridge estimate is the most popular one in the study of biased estimate. But it may not fully address the ill conditioning problem. So a new class of biased estimate was introduced by K.J. Liu in paper [35]. In this thesis, we did more research on LIU estimate and draw the related conclusions.In this thesis, we first presented the current research of parameter estimate. Then in chapter two, we introduced some basic knowledge of matrix, linear model, and biased estimate. In chapter three, we researched the advantages of general LIU estimate of Gauss-Markov model. In chapter four, restrict LIU estimate and general restrict LIU estimate of the linear model with equation restriction were introduced. And we discussed the properties and admissibility of both. Besides, we proved that restrict LIU estimate surpassed the least square estimate in the scalar Mean Squared Error and Mean Squared Error Matrix senses, and general restrict LIU estimate surpassed restrict LIU estimate in the scalar Mean Squared Error sense. In chapter five, we introduced a new class of principle component estimate: LIU-type principle component estimate, and proved that it surpassed the least square estimate in the scalar General Mean Squared Error sense under certain conditions. Besides, we proved that it surpassed the principle component estimate in the scalar Mean Squared Error sense. We also used the numerical example to illustrate the properties of LIU-type estimate.