A Study Of Computation And The Best Linear Unbiased Estimation Under The Singular Linear Model

Linear statistic model, which is abbreviated as linear model, is a very important type of statistic model. On the hand, this is because it includes a series of statistic models, such as linear regressive model, variance analysis, covariance analysis model, Variance component model, and etc. Those entire statistic models have been widely and extensively applied in the development of national economy. They all can be converted into the following general form of linear statistic model: Y = Xβ+ε . This model is usually abbreviated as (Y,Xβ,σ²V). The basic characteristic of the linear model is the unknown parameter of the model is linear and it also includes the linear random error.On the other hand, the research on linear model can provide the basic theoretical tools for other statistic technologies, such as experiment design, discriminate analysis, biology statistics, increasing curve analysis, multivariate statistical analysis, time sequential analysis and etc, therefore, linear model plays a very important role both theoretically and practically.The parameter estimation of the linear model is one of the basic issues of the linear model statistical deduction. Among which, least squares method is the most widely applied method of parameter estimation .If V > 0, the essential problem of parameter estimation is to work out the extreme small value of the quadratic function (Y-Xβ)′V^-1(Y-Xβ) for parameter β.But if V is singular, this method will not be able to applied. Actually this kind of singular linear model is much more frequently used in the practical application.This is the reason why the parameter estimation of singular linear model has always remained as a very active research field.In this paper, we presented an equivalent characterization of the best linear unbiased estimation, which has the same form as the optimal weighted least squares estimation, under the general linear model \Y,X/3,cr2V). An example demonstrates that the existence condition of a symmetric reflexive generalized inverse of Vcannot be reduced to the existence of a generalized inverse of V . In a lot of practical applications .especially the huge data processing of data mining , the applicable calculating method is the key point of whether there will be a successful conclusion for the theoretical result .The computation of best linear unbiased estimators on the singular linear model is mainly discussed. By the use of generalized inverse of partitioned matrix, a new expression of blue is given. The computation is smaller than C. R. Rao' s. To the nonnegative definite matrix in the new expression, we give the algorithm of generalized inverses of partitioned matrix and the algorithm of sweep operator.