Superiority And Optimal Prediction Of Biased Estimation

Linear model is an important statistical model, which has an extensive and important implications in many fields such as industry, economy, biology, medicine and so on. The basic theory and methods of linear model provide fundamental tools for other statistical theories and methods, in which parameter estimation and prediction are two significant problems in the research of linear model.The study on the problem of parameter estimation show that the classical least squares estimation is not a good estimation in many cases. If the problem that we concern is the error between the estimated value and real value but not the estimator class or methods for judging the superiorities of estimation, then the parameter estimation of linear model is not confined the least squares estimation. Therefore, many new estimations are proposed, where the biased estimation, i.e., the mean and parameter vector of estimation is not equal, is a very important one. In this paper, we study the superiorities of two biased estimations compared with the least squares estimation and the problem of determining the bias parameter further on the basis of the existing literatures.For linear model, we present a sufficient condition that the generalized ridge-type estimator is superior to the generalized least-squares estimator with respect to the mean squared error matrix criterion in terms of the ill-conditioned problems of design matrix and the ellipsoidal constraints on regression coefficient. The generalized ridge-type principal components estimator is an unified expression of common linear biased estimation. This paper discuss its admissibility and superiorities compared with least squares estimator, principal components estimator, generalized ridge estimator with respect to the mean squared error matrix criterion.On the other hand, the problem of prediction of model, i.e., predicting the unknown observations using the known ones, plays an important role in various fields. This paper, in prediction model, give the comparison of superiorities between the optimal observation and classical observation and obtains the necessary and sufficient condition of optimality of two kinds predictions in different rule. Furthermore, we also discuss the superiorities of two kinds predictions based on the biased estimation.The problem of determining the bias parameter is an important issue in practice. In this paper, aiming at the fact that the generalized ridge-type estimator is a self-adaptive nonlinear estimator, we propose concrete methods to determine ridge parameter in generalized ridge-type estimator using the linear Minimax estimator and balanced loss function. An example is also presented to analyze and compare above methods using R software.