Improved C-k Estimators Of The Parameters In Seemingly Unrelated Regression System

In this paper, we do some research on the system composed by two simple linearmodels which are only correlative between the errors -the seemingly unrelated regressionsystem, the basic model is :where yi is a n×1 vector, rk(Xi) = pi,βi is a pi×1 unknown regression parameter, ei isa n×1 random error vector, V = (σij) > 0.When the design matrix is ill-conditioned, the MSE of the covariance adjusted esti-matorβ-1 is very big, so the veracity ofβ-1 is deeply reduced as an estimator ofβ1. Wepropose a class of new estimators -the c - k improved estimatorsβ-1(c,k) to reduce theMSE.First, we prove thatβ-1(c,k)is more e-cient thanβ-1, the ridge improved estimatorsβ-1(k) and the Stein improved estimatorsβ-1(c) under some conditions; the admissibilityofβ-1(c,k) in the c - k improved estimators class, and the admissibility in the linearestimator class under some conditions; When V is unknown, the properties for the two-stage estimatorsβ-1(T,c,k) are given. Especially, we prove thatβ-1(T,c,k) is as e-cientasβ-1(c,k) for big sample size. Besides the properties are also discussed for the improvedestimators sequence.Second, based on the above model, when rk(X1) < p1,β1 is not always estimable,so we discuss the estimable functionαβ1, and prove thatαβˉ1(c,k) is more e-cientthanαβˉ1(1,0),αβˉ1(1,k) andαβˉ1(c,0) under some conditions. Especially, we get thenecessary and su-cient condition whichαβˉ1(c,k) is more e-cient thanαβˉ1(1,0); Theproperties of the two-stage estimators are studied at the same time.Third, we study the properties of the system which yi is ni×1 and n1 = n2, andprove thatβ¨1(T,c,k) is as e-cient asβ¨1(c,k), when the sample capacity is big.At last, under the seemingly unrelated mixed e-ect system, we get a better concomi-tant variable, and propose the c - k improved estimators with it.Then some statisticalproperties of the c - k improved estimators are studied as well.