The Relative Efficiency of the Conditional Root Square Estimation of Parameter in Inhomogeneous Equality Restricted Linear Model

This paper made a discuss on the relative efficiency of the generalized conditional root square estimation and the specific conditional root square estimation in paper [1,2] in inhomogeneous equality restricted linear model. It is shown that the generalized conditional root squares estimation has not smaller the relative efficiency than the specific conditional root square estimation, by a constraint condition in root squares parameter, we compare bounds of them, thus, choose appropriate squares parameter, the generalized conditional root square estimation has the good performance on mean squares error.

where Y is a n-dimension vector, X is a -order design matrix which is known, e is -random error vector, n p  1 n  n I is n-order unit matrix, is error variance, R is -matrix, r is dimension vector, is a n-dimension parameter vector which is unknown.This paper all assume X is full rank of column, R is full rank of row.
The RLSE of the regression coefficient  in the model ( 1) is noted as in the paper [1], where , . In the paper [1], the conditional root square estimation of parameter of restricted linear model is derived, when multicollinearity of explanatory variables exists.It is shown that it has smaller mean squares error than the RLSE, and the admissibility of the conditional root estimation is discussed.Under the MDE matrix comparisons criterion, the necessary and sufficient condition or sufficient condition, under which CRSE is superior to RLSE, is obtained.Two methods (Root Trace, Variance Inflation Factor) are used to evaluate the optimal value.In the paper [2], proposes generalized root squares estimation in inhomogeneous Equality Restricted Linear Model, we show that it have smaller mean squares error than the conditional root squares Estimation, and give display solution of generalize root squares estimation, propose the estimate methods of the optimal parameter value.Based on all the research work above, we made a discuss on the generalized conditional root square estimation and the specific conditional root square estimation in paper [1,2] that the relative efficiency has in inhomogeneous equality restricted linear model.It is shown that the generalized conditional root squares estimation has not smaller the relative efficiency than the specific conditional root square estimation, by a constraint condition in root squares parameter, we compare bounds of them, thus, choose appropriate squares parameter, the generalized conditional root square estimation has good nature on terms mean squares error.

Definition and Lemma
Definition 1 [1] In the model (1), defined as   k   is the specific conditional root square estimation of  : Non-zero characteristic values of W, and is the generalized conditional root square estimation of  :


(that efficiency highter),   improve the degree of  bigger. Lemma 1 is positive semidefinite matrix, and rank of W is .
Non-zero characteristic values of W, and  is Non-zero characteristic values of W, and 1 2 0 , , , , ,

Main Results
We can prove the following exist theorem and bound of and . Now, we have the following lemma.
accordingly, the specific conditional root square estima- similarly, the generalized conditional root square estima- , where Proof: So, there always exist 0 Lemma 10 In the model (1), exist root square parameter 0 < k < 1, then mean squares error of   Lemma 11 In the model (1), 1 Theorem 1 In the model (1), , always exist .
Proof: Based on the lemma 9 an 10.
Proof: Based lemma 11 and d 3, we get the usion.

efinition concl
Theorem 2 In the model (1), for , exist  , based on lemma 6 and, we have based on lemma 9, we have , then For above theorem, then 0 , he following the conclu . (1), Using theorem 2, we get t sion.Inference 1 In the model for , exist 0 2, e model ( 1), when we get the conclusion.
Theorem 4 In the model (1), for Proof: By theorems 3 and 4, we get 1 1 1 k p q k p q p q p q k p q p q p q p q p q k p q p q p q k p q p q p q p q p q k p q p q p q p q D 1 2 1 1 .
p q p q p q k p q p q p q p q b b b

5
In the model (1), assume the non- defined as above paper.