The Relative Efficiencies Of Mixed Estimator With Respect To Least Square Estimator In Linear Regression Model

The relative efficiencies of parameter estimators in linear model are the problems which we talk about often. Usually the parameter estimators have two kinds, one is the best linear unbiased estimator of the unknown parameter( BLUE ) ,and the other is the linear square estimator of the unknown parameter(LSE ) .First we consider the following models: y = Xβ+ e, E ( e ) = 0, Cov ( e )=σ2I, (1) y = Xβ+ e, E ( e ) = 0, Cov ( e )=σ2V. (2)In model (1), we know that the best linear unbiased estimator of function C′βis equal to its linear square estimator, but in many questions ,we can't think that the hypothesis of the error vectors is right. Because their error deviation may be not equal to each other, and maybe correlative to each other, at this time, its covariance is Cov ( e )=σ2V,and the model becomes model(2), in this model, when the matrix has complete rank,βis estimating, and its least square estimator isβ? = ( X′X )?1X′Y,its best linear unbiased estimator isβ* = ( X′V ?1 X )?1 X′V ?1Y,but in many theory and utility problems , V often includes unknown parameters .If we write the...