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

(1)

The Relative Efficiency of the Conditional Root Square

Estimation of Parameter in Inhomogeneous

Equality Restricted Linear Model

Xiuli Nong

Guangxi Normal University for Nationalities, Chongzuo, China Email: [email protected]

Received May 22,2012; revised June 30, 2012; accepted July 11, 2012

ABSTRACT

This paper made a discuss on the relative efficiency of the generalized conditional root square estimation and the spe-cific 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 condi-tional 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.

Keywords: Generalized Conditional Root Square Estimation; Specific Conditional Root Square Estimation; Relative Efficiency

Consider inhomogeneous equality restricted linear model

 

2

, 0, _n

Y X e E e Cov e I

R r

 

    

  



(1) where Y is a n-dimension vector, X is a -order de-sign matrix which is known, e is -random error vec-tor,

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.

2 ₀

 

q

ˆ  

p

 B



:Rr



q1

The RLSE of the regression coefficient  in the model (1) is noted as WX Y V in the paper [1],

1

1 



where W S1S R RS R1 _ _ RS1, SX X , 1

1 1

V S R RS R _  _ r. 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 suffi-cient condition or suffisuffi-cient 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 general-ize 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 condi-tional root square estimation and the specific condicondi-tional root square estimation in paper [1,2] that the relative ef-ficiency 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 com-pare bounds of them, thus, choose appropriate squares parameter, the generalized conditional root square esti-mation has good nature on terms mean squares error.

1. Definition and Lemma

Definition 1 [1] In the model (1), defined as 

 

k is the specific conditional root square estimation of  :

 

k

 

k





k W W WX Y V

_ _  __  _ _

k

, where 0 < k < 1,

 

diag



1 , , , 0, , 0



k k p q

W Q     Q,

W, V defined as above paper, Q is p-orthogonal matrix, make Q WQ diag



 1, 2,,p q , 0,, 0 ˆ



  , i is

Non-zero characteristic values of W, and

1 2 p q 0

(2)

Definition 2 [2] In the model (1), defined as 

 

K

is the generalized conditional root square estimation of

 :

 

K

 

K





K W W WX Y

_ _  __  _

V

 .

where Kdiag



k k1, 2,,kp



,

 



1 2



1 2

diag , , p q, 0, , 0

K _k

k k

p q

W Q      Q.

said k k1, 2,,kp q is -root square parameter, W,

Q, V defined as above paper.

p q

Definition 3 [3] Two estimation  and ˆ of the model (1), defined as

 

MSE MSE ˆ

, 1

ˆ

e   



  

 is elative

efficiency of estimation  for elative efficiency esti-mation of ˆ. If ˆ is the best linear unbiased estima-tion of  , then note e

 

 , ˆ e

 

 .

For the above definition 3, if , then

shows that

 

ˆ 0e  , 1

 is better than ˆ under mean squares error and if the bigger of e

 

 , ˆ (that efficiency highter),  improve the degree of ˆ bigger.

Lemma 1 [1] WSW W, WSW W.

Lemma 2 [1] 1 is posi-

tive semidefinite matrix, and rank of W is . 1

1 1 1

WS S R RS R _  _RS p q





Lemma 3 [1] Exist Q is p-order orthogonal matrix, make Q WQ diag



 1, 2,,p q , 0,, 0 ˆ



 , i is

Non-zero characteristic values of W, and

1 2 p q 0

  _ _  .

Lemma 4 [1] Mean squares error of  is

 

2

1 MSE

p q

i i

   





 , i is Non-zero characteristic val-

ues of W, and  1 2p q 0.

Lemma 5 [1] Assume Q V



b b1, 2, ,bp q , ,bp





  _ _

then



1 1, , p q p q, , p p



,

Q Q V

b b

 

  _ _ 

   

b 

  _  _ 

where p q 1p q 2p0.

2. Main Results

We can prove the following exist theorem

and bound of and . Now, we have the following lemma.

 







 



0e  k e  K 1

 





e  K

 e





 

k



Assume  Q



 V





 1, 2, ,p



, then  

   _

1 p q 2 p 0

p q

        .

And the RLSE of  is









Q V Q WX Y V V Q WX Y

Q WQQ X Y Q X Y

          

    

  

 

accordingly, the specific conditional root square estima-tion of  is

 



 



 





 

,

k

k k

k Q k V Q W WX Y V V

Q W WX Y Q W V Q V

Q W QQ WQQ X Y Q W QQ V Q V

Q X Y Q V Q V

I Q V

 





 

 

  

   _   _

 

   

  

      

 

   

     

 _{ }

   _  _

 

 





0 < k < 1.

similarly, the generalized conditional root square estima-tion of  is

 

K Q



 

K V

  

K

 

K I Q V,

 _ _  _ _{ } _{ }  _  _

 

 



 



1 2



diag , , , p

K k k  k .

Lemma 6



 



 

1 MSE

p q

i i

K g k

 











2





2

1 2

1

i i

k k

i bi i

 

 

, where

 

2

i i

g k      .

Proof:

when ki0,

 

2

1 1

0 MS

p q p q

i

i i

g   E 

 

 



 .

Because 

 

K  

 

 K _

 

 KI Q V _

 

 

 







 



, so







 



2

MSE  K tr cov  K  E  K  . Because

 









 



 



 

2

 

2 1 2

1 cov

cov

cov cov

i

K K

K K K

p q

K K _k

i i

tr K

tr I Q V

tr tr

tr W



 

  

 

 

 



  _ 

 _ _     _

 



  

  __  __  



   





  

So

 





 



 



 









2

2 2

1

i

k k

p q

k i i i i

E K E I Q V

I Q V E K

I Q V

b

 

  

 

 

 



 

  _{ } 

 _   _  _ _

 

 _{ }

   _  _ 

 

 _{ }

   _  _ 

 





 

 

(3)

Lemma 7 MSE





 

k



MSE





 

k



,

 







 



MSE  K MSE  K . Lemma 8 When0 1

2

k

  , exist 0 0 1 0

2

k _ k  _

 ,

when 0 k k0, then

 





₂





2

2 1 2

1

k k

g k    b  

has minimum value.

Proof: Note , , then

 

2 1 2

1

k

g k  



2





2

1

k

b

  _

 



2

g k    g k

 

g k1

 

g2

 

k .

For g k1

 

, we have

 

2 1 2

1 2 l

k

g k     n. When 1

0 2

k

  , if 1, then ln0, 1 2k 1; if 0  1, then ln0, 1 2 k 1. When 1, g k₁

 

0. So

 

1

g k is a monotonically decreasing function in 0,1 2

 

 

 . For g2

 

k , when 1, we have

 





2





2 2 1 l

k k

g k   b    n0. This means

 

2

g k is a monotonically increasing function in 0,1 2

 

 

 . So, there always exist 0 0

1 0

2

k _ k  _

 , when 0 k k0, we have g k

 

g k1

 

g2

 

k 0 , so g k

 



0



is a monotonically decreasing function in 0,k , g k

 

<

,

 

0

g g k

 

has minimum value. Lemma 9 In the model (1), for

 

2 1 2





2



2

1

k

g k    b  









k



, when

2 2

2 ln

ln

b b k

  

 

 



 , then g k

 

has minimum value. Proof: According to lemma 8,

 









1 2

2 2 1 2

2 kln 2 k 1 ln

g k g k g k

b k .

     

    

      

Let g k

 

0, we get





2





2 1 2

2 kln 2 k 1 ln

b

      

    k _0

, when  1, the solution of this equation is k0; when  1, the solution of this equation is









2

2 2

k b

b  

  

 _ 

 









2 2

2



en

0, 1

ln

, 1

ln

b k

b



 



 

 



  

 

 

 

. Therefore wh So









2 2

2 ln

ln

b b k

  

 

 



 , then g k

 

has minimum value. Lemma 10 In the model (1), exist root square pa-rameter 0 < k < 1, then mean squares error of 

 

k is

 





2



MSE

p q

k k

i i i i

k b

  2 1



2





2

1

   

1

i

 _ _ 

 _  _

 



 .

Lemma 11 In the model (1), 1

p q

i 







, a exist 1

i

lways 1

0 2

k

  , then MSE





 

k



MSE







d lemma

Theorem 1 In the model (1), , always exist .

Proof: Based on the lemma 9 an 10.

1 1

p q

i i











1 0

2

k

  , then 0e





 

k



1.

Proof: Based lemma 11 and d 3, we get the usion.

efinition concl

Theorem 2 In the model (1), for k0 , exist



1



diag , , p

K k  k , then



1

Proof: For

 

0e



 K ,

 

k  .

, that









2 2

2 ln

ln

b b k

  

 

 



 .

0

k , if









2 2

b

2 ln

b k

ln

i i

   

 



 ,

1, 2, ,

i _ p, choose kik, we have

   



, k



0

e  K   .









2 ln

2

ln

i i

b

b k



Assume at least exist i, that

  



 



 ,

assume









2 2

1 1 2 1 1 1

ln

k

ln b

b

  



 



where



 , k_ik,

2, 3, ,

i _ p, based on lemma 6 and, we have

 







 



MSE MSE

,

p q p q i

k K

 

1

 

1

1 1

i i i

g k g k g k g k







 

 

 









 

based on lemma 9, we have g k

 

g k1

 

1 0, then

 







 



MSE  k MSE  K 0

   

, so



,



0

(4)

For above theorem, then 0e





   

K , k



1 he following the conclu

. (1),

Using theorem 2, we get t sion. Inference 1 In the model for k0 , exist



1



diag , , p

K k  k , then 0e





 

k

 

e 

 

K



1

(1), if re

no

Proof: Because Q is orthogonal m trix, so when

. Inference 2 In the model _i



i1, 2,_,p



A t all equal, then 0e





 

k



e





 

K



1.





Q

   V ,

a-1 2

  , then 12, based on theorem 2,

e model (1), when we get the conclusion.

Theorem 3 In th









2 ₂ 2 ln 0 p q p q k          1 ln

p q p q p q b b  _ _  _       , then e





 

k



e00, where





2





2

2 2 1 2

1

0 2

1

k k

p q p q bp q p q

e      

 

 

       

 ,

p q

 _ is  Q



V



the largest component of mod-ule.

Proof: Assume  2 1



p q bp q



2, then

 





























2 2

2 2 1 2k k

    

    

1 1

2

i i

 2 

2 1 2

2 1 2 2 2 2 2 1 MSE MS 1 1 1 1 .

p q p q

i i i i i

k p q

i i i k

i i i

k p q

i i i k i i i i b b b                                            _ _          



Assume E k  





2





2

2 2 1 2

1

0 2

1

k k

p q p q bp q p q

e      

             , then

 





2

0 0

1

MSE MSE MSE

p q

i i

k e e

    



 





  ,

so



 





 



 

0

MSE

1 0

MSE

k

e  k  e



    



 .

Theorem 4 In the model (1), for Kdiag



k₁,_,k_p



,

if









2 ₂ 2 ln ln i i i i i b b k         









2 2 2

or ki i i

i i i

b b        _   _ _      i ,

1, 2, ,p

 _ , then e





 

K



 e1 0,



Where



2

1 p q bp q



      



1 2

e  .

Proof:













































2 2 2 1 1 2 2 2 4 2 2 2 2 2 2 1 1 1 1 i i i i p q k k

i i i i i

k k

i i i i i

K b b b b b b                           _ _          _       _  _  _ _   _ _      _ _  _{ }  _ _ _     









2 1 2 2 2    



1 2 1 2 2 MSE MS 1 p q i i p q i p q i i i i i b                         



E 





























2 2

4 2 2 4 2

2 2

2

4 2 4

2 2 2

2 1 1 2 2 2 2 1 2

i i i i i i i i i i i i i

p q

i i i i i

i i _i _i _i

i i i

i i i p q

i i p q p q

b b b b b b b b b                                   1 6 3 2 2 2 1 2 2 1 2 p q i p q p q i i                          _   _ _ _ _       _ _ _ _           _ _       







  



2MSE .

p q bp q

      2 1  

Therefore



 







2 2 2 1 0

p q p q

e K b      _ _     





, let 2

1 ₂ 2

1 p q p q

e

b



   _ _



  , then

Theorem 5 In the model (1), assume the non-zero ch root

 





1 0

e  K  e .

aracteristic i of W are not all equal



i1, 2,_,p



, for the efficiency lower bound e0 of

 

k

 and the efficiency lower bound e1 of 

 

K , the relationship of them is e0e1.

Proof: By theorems 3 and 4, we get





2





2

2 2 1 2

2 1

1

k k

p q p q bp q p q

e     

(5)

, also 1 2 k0 1 p q

  _ , then 11 2 0, thus

k p q

 



 

1 2

D D 0

That .





2

1 2

e

b



  



  2,

note

0 1

e e . 1 p q p q

1 1

0 2

e D

e D then

4

1 1

D   ,





















2 2

4 4 2 2

2 1 1

2 2

2 2 1 2

1 2

1

k k

p q p q p q p q

k

p q p q p q p q p q

k

p q p q p q

D b

b b

b

       

     

 

 

  



    



  

   

   

  4





2 1

  

Then

k











































1 2

4 4 4 2 2

1 1 1

2 2

2 1

2 ₂ _{1 2} 2

1

2 4

2

4 2 1 2 2

1 1 1

2

2 2

1

2

2 ₂

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 D

b

b b

b

      

   

   

 

      

    

    

 



  



    



  



 

  



   



  

  

  

  

 

  _   _

 

 

 _   _

 

   





















2

4 2

1 1

2 2

2 1 2

1 1 .

p q

p q p q k

p q p q p q p q

b

    

    



 



   

REFE

and W.-R. L Square Estimation of Parame

Journal of Chongqing e Edition),

Liu, tional Root Squares Estimation

Journal of Gua

No. 3, 2007, pp. , “The Relativ

ation,” Journal o

, pp. 13-15.

2

 



 _ 

 

 

 

 _   _

 



_     _

are not all equal



i1, 2,_,p



, therefore As i

RE

iu, l Root

(Na-

ienc No. 2, 20

X.-L. Nong, W.-R. et al.,

o stricted

of

Tech-, Vol. 18Tech-,

ang e Effi ized

f Q College, 1998

NCES

“The Conditiona ter of Restricted Linear

Normal University

07, pp. 24-28.

“The Generalized Condi-f Parameter in Re

ngxi University

24-27.

ciency of the General

uanZhou Normal

[1]

Model,”

tural Sc

[2]

nology

[3] P.-H. W

Vol. 2 X.-L. Nong

Linear Model,”