Theory And Methods Of Parameter Estimation In Linear Models

The thesis is concerned with the estimation theory, methods and some otherrelated problems of the parameters in linear models.For the balanced variance components mixed models, we propose a new esti-mate, called the spectral decomposition estimate, and study the spectral decom-position of covariance. This estimation method can give the estimates of the fixede?ect and the variance components simultaneously. Furthermore the nonnegativeof the covariance matrix which is gotten by the spectral decomposition estimatehas always been guaranteed. By using the layer sorting introduced in this thesis,we can obtain the spectral decomposition estimate of the covariance matrix simplyand conveniently, and we can determine directly the number of distinct eigenvaluesof covariance matrix. In particular, we first explicitly give the expression of theeigenvalues by using spectral decomposition estimate method, when the numbersof distinct eigenvalues are greater than those of variance components. In addi-tion, we can obtain the estimates of the variance components, and the calculatedproceed only uses the original relation matrix and its converse matrix. We alsostudy the properties of the spectral decomposition estimate and the risk functionunder the matrix squared error loss. It is proved that under some mild conditions,three estimates (the spectral decomposition estimate, analysis of variance estimateand minimum norm quadratic unbiased estimate) of the covariance matrix havethe same risk function, in particular, which has always holds for random e?ectsmodels.When an independent estimate of covariance matrix is available, we oftenprefer to two-stage estimate (TSE), we study the expressions and properties ofthe two-stage covariance adjusted estimate and the selection of covariables, andpropose the method to select the high correlative covariables. Expressions of theTSE's covariance matrix obtained by using all or some covariables in covarianceadjustment approach are given. We propose a necessary and su?cient conditionthat the TSE is superior to the least square estimate (LSE) and the related largesample test is also established. Furthermore the TSE by using some covariables isexpressed as weighted least square estimate. Based on this fact, a necessary andsu?cient condition that the TSE by using some covariables is superior to the oneby using all covariables is obtained. These results put some insight on selection ofcovariables in the TSE and its application. Under three covariance matrices whichare usually supposed in economics, biology, medicine and other fields, we give thesimulation results. From those ones, we can see that the two-stage covarianceadjusted estimate have a smaller mean squared error after selection, those implythat it is necessary to select covariables and canonical correlation test we proposedhas excellent property.The problem fitting a circle data has often occurred in many engineering andtechnology fields. A statistical treatment using a linear model with heteroscedas-...