Study On The Singular Lnear Model Of Biased Estimation And The Factor Analysis Model

Matrix knowledge, especially the generalized inverse matrix is vital for the study of linear model. In this paper, we firstly studied the various structures and properties of matrix left symmetry factor, right symmetry factor and symmetry factor from the algebra point of view. Then, we proposed the construction of {2, 3}-inverse and {2, 4}-inverse with prescribed range and null space based on the research on {2}-inverse, which suggests a new way for studying the generalized inverses.It's known that the biased estimation in linear model has received much attention all the time in the regression analysis in statistics. The biased estimation is the most direct method in improving the OLSE with the existence of multicollinearity problem. However, most existing literatures only deal with the case when the covariance matrix is positive definite. In this paper, we proposed some new insights into the unified theory of least squares in order to study the singular linear model. Applying the unified least squares theory, we know that it's easily to get the corresponding results in the singular case with the given results in the non-singular linear model in many situations. But it's not always true since the matrix T may be singular, so we discuss the ridge type unified least squares theory, which is an alternative method to the unified least squares theory in dealing with such problems. After that, we discussed the parameter estimation in the linear model with the ellipsoidal restriction in the present of multicollinearty. The explicit estimator in such situation is obtained and its statistical properties are also studied. Furthermore, the unified biased estimator in the singular linear model and its properties are also derived in this paper.Lastly, we have also discussed the factor analysis model, and found that it's not necessary to standardize the sample data before being processed. Applying this model, we obtained 3 common explanatory factors based on the model analysis of the third quarter's financial report forms of 50 listed companies in 2005, and the comprehensive evaluation about these stocks are also made according to the factor scores.