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<strong>Ozean</strong> Journal of Applied Sciences 4(4), 2011<br />
E ( ˆ )<br />
2<br />
Var(<br />
ˆ) <br />
<br />
<br />
<br />
<br />
<br />
<br />
( ˆ )<br />
Var ( ˆ) <br />
2<br />
2<br />
1 <br />
( ˆ )<br />
Exp <br />
Var(<br />
ˆ) <br />
2Var(<br />
ˆ) <br />
E ˆ 2<br />
2 1 1 Z<br />
2 / 2<br />
( ) Var(<br />
ˆ) Z<br />
e Var(<br />
ˆ ) dZ<br />
2<br />
Var(<br />
ˆ) <br />
E(<br />
ˆ )<br />
2<br />
Var(<br />
ˆ) <br />
<br />
<br />
<br />
Z<br />
2<br />
1<br />
e<br />
2<br />
Z<br />
2 / 2<br />
dZ<br />
ˆ 2<br />
2<br />
E(<br />
)<br />
Var ( ˆ) E(<br />
Z ) Var(<br />
ˆ) <br />
~<br />
2 2<br />
MSE(<br />
) ( K 1)<br />
Var(<br />
ˆ) 2K(<br />
K 1)(<br />
Zero)<br />
K<br />
~<br />
2 2<br />
2<br />
MSE(<br />
) ( K 1)<br />
Var(<br />
ˆ) K Var(<br />
ˆ) <br />
~<br />
2 2 2<br />
MSE( ) Var(<br />
ˆ) ( K 1)<br />
K<br />
… (28)<br />
<br />
<br />
2<br />
Var(<br />
ˆ) <br />
… (29)<br />
2<br />
<br />
d ˆ <br />
<br />
By the same way the mean error squares for estimator shrinkage ( ~ ) can be yielded by the following function:<br />
MSE( ~ )<br />
K<br />
<br />
2 2 2<br />
Var( ~ ) ( K 1)<br />
<br />
… (30)<br />
<br />
The relative efficient for the estimated ~ and ~ for the estimators ˆ and ˆ which can be calculated by<br />
Restricted Least Square method (RLS):<br />
~ MSE( ˆ) <br />
R.<br />
E(<br />
) ~<br />
MSE(<br />
)<br />
~ Var(<br />
ˆ) <br />
R.<br />
E(<br />
) <br />
2 2<br />
Var(<br />
ˆ) (<br />
K 1)<br />
K<br />
~<br />
2 2 2 <br />
R.<br />
E(<br />
) (<br />
K 1)<br />
K 1<br />
( ˆ)<br />
( ~ MSE <br />
R.<br />
E ) <br />
MSE( ~ )<br />
( ˆ)<br />
( ~ Var <br />
R.<br />
E ) <br />
2 2<br />
Var( ~ ) ( K 1)<br />
K<br />
R.<br />
E( ~ ) <br />
2<br />
<br />
… (31)<br />
<br />
2 2 2 <br />
(<br />
K 1)<br />
K 1<br />
2<br />
<br />
… (32)<br />
To calculate K value that make ( )<br />
squares calculated for K and equaling it to 0:<br />
~<br />
MSE and (~<br />
)<br />
MSE the least, the first derivative for the mean error<br />
~<br />
MSE( )<br />
2<br />
2( K 1)<br />
Var(<br />
ˆ) 2K Var(<br />
ˆ) <br />
K<br />
~<br />
MSE(<br />
)<br />
0<br />
K<br />
401
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