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C03ed19.doc 7/8/2003 4:11 PM 1
Special Cases of the MSE of Predictors (corrected version of c02ed30.doc)
Ed Stanek Introduction We consider special cases of the MSE of predictors, using the results in c03ed18.doc. The results are first given, followed by special cases. Basic Results
We summarize the results for the MSE. First, the predictor of
1
N
is ss
T U µ=
′= =∑g Y (1.1)
is *
1 1, 2 3ˆ
I II II IT ′ ′ ′ ′ ′ ′ + + g C g C g C Y= (1.2)
where I I′ ′ ′=g g K ; 1, 1,II II′ ′ ′=g g K ; 2, 2,II II′ ′ ′=g g K ; ( )1, 2,II II II′ ′ ′=g g g ;
( ) ( )I n mn N n m M m× − × −
= ⊗
K I 0 I 0 ; ( ) ( )1,II n M mn N n M m m −× − − ×
= ⊗
K I 0 0 I ;
( )2,II N n MN n n −− ×
= ⊗
K 0 I I ; ( ) *1
m nme n m r nk k
m nm ′ = ⊗ + ⊗ +
J JC I P P ;
( )*2
M m mnk
m− ×
′ = ⊗
JC P ; ( )
3NM nm nm
nm− ×′ =J
C ; 2
2 2e
ee
kσ
σ σ=
+;
( )*2 2
**2 2 2
er
e
mk
mσ σ
σ σ σ+
=+ +
and
*2*
*2 2 2e
mkm
σσ σ σ
=+ +
. The MSE is given by
( )( )( )( )
( )( ) ( )
( )
1 2 3
21, 2 2 1,
23 , 2 1,
23 3 , 3
21 1 1 1
21, 2 1 2 1
ˆvar 2
2
2
2
II I nm II
II I nm II I II
II I nm II I II II
I nm I nm I
II I nm I
II
Tξ ξ ξ
σ
σ
σ
σ
σ
′ ′ + ′ ′= + + − + ′ ′+ + − +
′ ′ ′ − − + ′ ′ ′ + + − +
′ ′+
g C V I C g
g C V I V C g
g C V I C V C V g
g C I V C I C C g
g C V C I C C g
g ( )( ) 23 , 1 3 1I II I nm Iσ
′− − + C V V C I C C g
. We re-express one of the terms in the MSE. The term is given by
C03ed19.doc 7/8/2003 4:11 PM 2
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
2 2
2 2
2 2 2 22 2
1
1
1 1
e
n M m NM nm
e
N n M NM nm
e e e e
n M m n M m n M m N n
mn nm M nm
mn nm M nm
n M nm nm n M nm nm
σσ σ
σσ σ
σ σ σ σσ σσ σ
2
− × −
2
− × −
2 2
− × − − × −
− + + +
= + − +
− − + + − − + +
=
J
J
J J
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2 2 2 22 2
2 2 2
1 1
1
M
e e e e
N n M n M m N n M N n M
n M m n M m n M m N n Me e
NM nm
N n M n M m N n M N n M
n M nm nm n M nm nm
nm n M
σ σ σ σσ σσ σ
σ σ σσ
2 2
− × − − × −
− × − − × −2
−
− × − − × −
− + + − + +
− − + = + −
J J
J JJ J J
As a result, we can express the MSE as
( )
( )( )
( )( ) ( )
( ) ( )( )
1 2 3
2 22*2 2 *
1, 1, 1,
2
ˆvar
2n M m
e eII n M m II II II
N n M nm
M mn M m n M mn M m N n M
II II e IIN n MN n M n M m
T
M mk kM m m M
mM M m
ξ ξ ξ
σ σσσ σ
σ σ
−2
−− ×
−− −− × −2
−− × −
=
⊗ − ′ ′+ + ⊗ − −
⊗ ⊗ + −′ ′+ + ⊗
P Jg P J g g g0
JI J 0 I Pg g g
0 I J( ) ( )
( ) ( )( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2 2 2
2
1
n M m N n MII
N n MN n M n M m
n M m n M m n M m N n Me e
II NM nm II II II
N n M n M m N n M N n M
I e
nm n M
k
σ σ σσ
σ
− × −
−− × −
− × − − × −2
−
− × − − × −
+ ⊗
− − + ′ ′+ + −
′
+
0g
0 I P
J Jg J g g gJ J
g ( ) ( ) ( )( )
( ) ( )
( )
( )
( )
* * * * *2
* * * *1,
*
1 1 1
2 1
2
m nmn m r r e r n I
M m mr e II n I
n M m m NM nm nmII I
N n M nm
k k k k k km nm
k k k k km
km nm
− ×
− × − ×
− ×
⊗ + − + − − + ⊗ +
′+ − − − ⊗
⊗ ′+ +
J JI P P g
Jg P g
P J Jg g0
. The predictor of AT given by
C03ed19.doc 7/8/2003 4:11 PM 3
( )*
1
N
A is s ss
T U Wµ=
′= = +∑g Y (1.3)
where 1
1 M
s stkt
W WM =
= ∑ is given by
*
1 1, 2 3A I A II II IT ′ ′ ′ ′ ′ ′ + + g C g C g C Y=
where 1A nm′ =C I . The MSE is given by
( )( )( )( )
( )1 2 3
21, 2 2 1,
23 , 2 1,
23 3 , 3
2
ˆvar 2
2
II I nm II
A II I nm II I II
II I nm II I II II
II II
Tξ ξ ξ
σ
σ
σ
σ
′ ′ + ′ ′= + + − + ′ ′+ + − +
′ +
g C V I C g
g C V I V C g
g C V I C V C V g
g g
or
( )
( )( )
( )( ) ( )
( ) ( )( )
1 2 3
2 22*2 2 *
1, 1, 1,
2
ˆvar
2
A
n M me e
II n M m II II IIN n M nm
M mn M m n M mn M m N n M
II II e IIN n MN n M n M m
T
M mk kM m m M
mM M m
ξ ξ ξ
σ σσσ σ
σ σ
−2
−− ×
−− −− × −2
−− × −
=
⊗ − ′ ′+ + ⊗ − −
⊗ ⊗ + −′ ′+ + ⊗
P Jg P J g g g0
JI J 0 I Pg g g
0 I J( ) ( )
( ) ( )( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2 2 2
2
1
n M m N n MII
N n MN n M n M m
n M m n M m n M m N n Me e
II NM nm II II II
N n M n M m N n M N n M
II
nm n Mσ σ σ
σ
σ
− × −
−− × −
− × − − × −2
−
− × − − × −
+ ⊗
− − + ′ ′+ + −
′+
0g
0 I P
J Jg J g g gJ J
g II g. Alternative expressions for the Common MSE Term We explore some alternative expressions for the common MSE term. Note that
( )1, 2,II II II′ ′ ′=g g g . As a result, the common term is just a function of three terms. The coefficients of these terms gives rise to the complications in the expressions. We simplify this by expressing
C03ed19.doc 7/8/2003 4:11 PM 4
( )( )
( )( ) ( )
( ) ( )( )
2 22*2 2 *
1, 1, 1,
2
2n M m
e eII n M m II II II
N n M nm
M mn M m n M mn M m N n M
II II e IIN n MN n M n M m
M mk kM m m M
mM M m
σ σσσ σ
σ σ
−2
−− ×
−− −− × −2
−− × −
⊗ − ′ ′+ + ⊗ − −
⊗ ⊗ + − ′ ′ + + ⊗
P Jg P J g g g0
JI J 0 I Pg g g
0 I J( ) ( )
( ) ( )( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2 2 2
1, 1, 1,
1
n M m N n MII
N n MN n M n M m
n M m n M m n M m N n Me e
II NM nm II II II
N n M n M m N n M N n M
II II I
nm n M
a b
σ σ σσ
− × −
−− × −
− × − − × −2
−
− × − − × −
⊗
− − + ′ ′+ + −
′ ′+
=
0g
0 I P
J Jg J g g gJ J
g Ag g( )( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
2, 1,
1, 1,1, 2, 1, 2,
2, 2,
1, 2,
I II IIN n M nm
n M m N n M n M m N n MII IIII II II II
II IIN n M n M m N n M n M m
n M m N n M
II II
N n M n M m
c d
e
− ×
− × − − × −
− × − − × −
− × −
− × −
′
′ ′ ′ ′+ +
′ ′+
Bg g0
C 0 E 0g gg g g g0 D 0 Fg g
G Hg g K L ( ) ( ) ( )
( ) ( )
1, 1,1, 2,
2, 2,
n M m N n MII IIII II
II IIN n M n M m
f− × −
− × −
− − ′ ′+
G Hg gg g K Lg g
where 2 2
*2 2 eM ma kM m m
σ σσ − = + +
,
2*2 eb k
Mσ
σ 2 = − −
, c σ 2= , 2
ed σ= ,
2 2eenm
σ σ+= , and
21 efn M
σσ 2
= −
. Also, n M m−= ⊗A P J , n M m−= ⊗B P J ,
n M m−= ⊗C I J , N n M−= ⊗D I J , M mn M m
mM M m
−−
= ⊗ + −
JE I P , N n M−= ⊗F I P ,
( ) ( )n M m n M m− × −=G J ,
( ) ( )n M m N n M− × −=H J ,
( ) ( )N n M n M m− × −=K J , and
( ) ( )N n M N n M− × −=L J .
Expanding these terms, the
C03ed19.doc 7/8/2003 4:11 PM 5
( )( )
( ) ( ) ( )
( ) ( )( ) ( ) ( )
( ) ( )
( )
1, 1, 1, 2, 1,
1, 1,1, 2, 1, 2,
2, 2,
1, 2,
II II II II IIN n M nm
n M m N n M n M m N n MII IIII II II II
II IIN n M n M m N n M n M m
n M
II II
a b
c d
e
− ×
− × − − × −
− × − − × −
−
′ ′ ′+
′ ′ ′ ′+ +
′ ′+
Bg Ag g g g0
C 0 E 0g gg g g g0 D 0 Fg g
G Hg g
( ) ( )
( ) ( )( ) ( ) ( )
( ) ( )
[ ] [ ][ ] [ ]
1, 1,1, 2,
2, 2,
1, 1, 1, 1,
1, 1, 2, 2, 1,
m N n M n M m N n MII IIII II
II IIN n M n M m N n M n M m
II II II II
II II II II I
f
a b
c c
× − − × −
− × − − × −
= − − ′ ′+
′ ′+
′ ′ ′+ + +=
G Hg gg gK L K Lg g
g A g g B g
g C g g D g g [ ] [ ][ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]
1, 2, 2,
1, 1, 2, 1, 1, 2, 2, 2,
1, 1, 2, 1, 1, 2, 2, 2,
I II II II
II II II II II II II II
II II II II II II II II
d d
e e e e
f f f f
′+ ′ ′ ′ ′+ + + + ′ ′ ′ ′+ − + + − +
E g g F g
g G g g K g g H g g L g
g G g g K g g H g g L g which we express upon grouping terms as
[ ] [ ][ ] [ ] [ ] [ ][ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]
1, 1, 1, 1,
1, 1, 2, 2, 1, 1, 2, 2,
1, 1, 2, 1, 1, 2, 2, 2,
1, 1, 2, 1, 1, 2, 2, 2,
II II II II
II II II II II II II II
II II II II II II II II
II II II II II II II II
a b
c c d d
e e e e
f f f f
′ ′ +
′ ′ ′ ′+ + + +
′ ′ ′ ′+ + + +
′ ′ ′ ′+ − + + − +
g A g g B g
g C g g D g g E g g F g
g G g g K g g H g g L g
g G g g K g g H g g L g
[ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ][ ] [ ] [ ]
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
2, 2, 2, 2, 2, 2, 2, 2,
2, 1, 1, 2, 2, 1, 1,
II II II II II II II II II II II II
II II II II II II II II
II II II II II II
a b c d e f
c d e f
e e f
=
′ ′ ′ ′ ′ ′= + + + + + −
′ ′ ′ ′+ + + +
′ ′ ′ ′+ + + +
g A g g B g g C g g E g g G g g G g
g D g g F g g L g g L g
g K g g H g g K g g [ ]( )
( )( ) ( )
2,
1, 1,
2, 2,
1, 2,
II II
II II
II II
II II
f
a b c d e f
c d e f
e f e f
−
′= + + + + − ′+ + + + ′ ′+ + + −
H g
g A B C E G g
g D F L g
g K H g. We simplify each of these terms. First, we simplify the expression for
( )a b c d e f+ + + + −A B C E G by replacing the matrices by their corresponding values. Hence,
C03ed19.doc 7/8/2003 4:11 PM 6
( )
( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( ) ( )
M mn M m n M m n M m n M m n M m n M m
n M m n M m n M m n M m n M m n M m
n M m n M m n M m
a b c d e f
ma b c d e fM M m
ma b c d d e fM M m
ma b c d d eM M m
−− − − − − × −
− − − − − × −
− − −
+ + + + − =
⊗ + ⊗ + ⊗ + ⊗ + + − = −
= + ⊗ + ⊗ + ⊗ + ⊗ + − =−
= + ⊗ + + ⊗ + ⊗ + − −
A B C E G
JP J P J I J I P J
P J I J I P I J J
P J I J I P ( )( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( )( )
( ) ( ) ( ) ( ) ( )
n M m n M m
n M m n M m
n M m
n M m n M m n M m
n M m n M m n M m
f
a ba b
nmc d
M M m
dd e fM m
a bm dd a b c d e fM M m M m n
− × −
− −
−
− − −
− − −
=
+= + ⊗ − ⊗
+ ⊗ + −
⊗ − ⊗ + − ⊗ =− +
= ⊗ + + + + − ⊗ + − − ⊗ − −
J
I J J J
I J
I I I J J J
I I I J J J
We can express a matrix of the form ( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( )
s t s t s t
s t s t s t s t
s t
s t s t s t
x y z
x x y xx yt t s st
y xzs st
x y xx y zt s st
⊗ + ⊗ + ⊗ =
= ⊗ − ⊗ + + ⊗ − + ⊗ + + + ⊗ =
= ⊗ + + ⊗ + + + ⊗
I I I J J J
I I I J I J J J
J J
I P P J J J
. (1.4)
Using this expression,
C03ed19.doc 7/8/2003 4:11 PM 7
( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( )( ) ( )
( )
1 1
n M m n M m n M m
n M m n M m
n M m
n M m
a b c d e f
a bm dd a b c d e fM M m M m n
m d dd a b c dM M m M m M m
a b m d de f a b c dn n M M m M m n M m
d a b
− − −
− −
−
−
+ + + + − =
+ = ⊗ + + + + − ⊗ + − − ⊗ − −
= ⊗ + + + + − + ⊗ − − − +
+ − − + + + + − + ⊗ − − −
= ⊗ + +
A B C E G
I I I J J J
I P P J
J J
I P( ) ( )
( )( ) ( )
( ) ( ) ( ) ( ) ( )
1
1
n M m
n M m
n M m n M m n M m
mc dM M m
a b me f a b c dn n M M m
m md a b c d e f c dM M m n M M m
−
−
− − −
+ + ⊗ −
++ − − + + + + ⊗ −
= ⊗ + + + + ⊗ + − + + ⊗ − −
P J
J J
I P P J J J
. Next, we simplify ( )c d e f+ + +D F L . Thus,
( )( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( )( )
N n M N n M N n M N n M
N n M N n M N n M N n M N n M
N n M N n M N n M
c d e f
c d e f
dc d e fM
dd c e fM
− − − × −
− − − − × −
− − −
+ + + =
= ⊗ + ⊗ + + =
= ⊗ + ⊗ − ⊗ + +
= ⊗ + − ⊗ + + ⊗
D F L
I J I P J
I J I I I J J
I I I J J J
.
Then, using the result in (1.4), ( )
( ) ( ) ( )( )
( ) ( ) ( ) ( )1
N n M N n M N n M
N n M N n M N n M
c d e f
dd c e fM
d d d dd c e f cM M N n M N n M
− − −
− − −
+ + + =
= ⊗ + − ⊗ + + ⊗
= ⊗ + − + ⊗ + + + − + ⊗ − −
D F L
I I I J J J
I P P J J J
which simplifies to
( ) ( ) ( ) ( )N n M N n M N n Mcc d e f d c e f
N n− − − + + + = ⊗ + ⊗ + + + ⊗ −
D F L I P P J J J .
Finally, we simplify the expression for ( ) ( )e f e f′+ + −K H such that
( ) ( ) ( )( ) ( )
( )( ) ( )
( ) ( )2
n M m N n M n M m N n M
n M m N n M
e f e f e f e f
e− × − − × −
− × −
′+ + − = + + −
=
K H J J
J.
C03ed19.doc 7/8/2003 4:11 PM 8
We summarize the results for the common terms, such that ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
1, 1, 2, 2,
1, 2,
1, 1,
2,
1
II II II II
II II
n M m n M m
II II
n M m
II N n
a b c d e f c d e f
e f e f
md a b c dM M m
me f c dn M M m
d
− −
−
−
′ ′+ + + + − + + + + ′ ′+ + + − =
⊗ + + + + ⊗ − ′= + − + + ⊗ −
′+ ⊗
g A B C E G g g D F L g
g K H g
I P P J
g g
J J
g I P( ) ( ) ( )
( ) ( )
2,
1, 2,2
M N n M N n M II
II IIn M m N n M
cc e fN n
e
− −
− × −
+ ⊗ + + + ⊗ − ′+
P J J J g
g J g
We now simplify terms in this expression. First, note that
( )2 22 *2
*2 2 *2 2 2 2e ee
M m ka k kM m m M m
σ σσσ σ σ σ − = + + = − + +
,
2*2 eb k
Mσ
σ 2 = − −
, 2c σ= , 2
ed σ= , ( )2 21ee
nmσ σ= + , and
21 efn M
σσ 2
= −
. We
represent 2
2 enfMσ
σ
= −
and ( )2 2enme σ σ= + . Then *2 *2a k nf k ne= + , *2b k nf= − ,
2c σ= , 2ed σ= , ( )2 21
eenm
σ σ= + , and 21 ef
n Mσ
σ 2 = −
so that all terms are given in
terms of c, d, e, and f. Substituting these equivalences into the expression above,
( ) ( )( ) ( )
( ) ( ) ( )
( ) ( )
1, 1, 2, 2,
1, 2,
*2 *2 *
1, 1,
2
1
II II II II
II II
n M m n M m
II II
n M m
a b c d e f c d e f
e f e f
md k nf k ne k nf c dM M m
me f c dn M M m
− −
−
′ ′+ + + + − + + + + ′ ′+ + + − =
⊗ + + − + + ⊗ − ′= + − + + ⊗ −
g A B C E G g g D F L g
g K H g
I P P J
g g
J J
( ) ( ) ( )
( ) ( )
2, 2,
1, 2,2
II N n M N n M N n M II
II IIn M m N n M
cd c e fN n
e
− − −
− × −
′+ ⊗ + ⊗ + + + ⊗ − ′+
g I P P J J J g
g J g
.
Now
C03ed19.doc 7/8/2003 4:11 PM 9
( )
( ) ( )
( )
*2 *2 *
2 2*2 *2 2 2 * 2
2 2 22*2 * 2
2
1 1 12
2
e ee e
e e ee
mk nf k ne k nf c dM M m
mk n k n k nn M nm n M M M m
mk km M m M M M m
σ σσ σ σ σ σ σ
σ σ σσσ σ σ σ
2 2 2
2 2 2
+ − + + =−
= − + + − − + + = −
= + − + − − + + −
Now
( ) ( )( )( ) ( )
2 22 2
22 2
2 2
e ee e
ee e
e e
m mM M m M M M M m
M m mM M M m M M m
M M m
σ σσ σ
σσ σ
σ σ
= − + +− −
−= − + +
− −
= − +−
.
Then
( )
( )
( )
*2 *2 *
2 2 22*2 * 2
2 2 2 2 22*2 *2 *
2 2 22*2 * *2
2
2
2
2 1
e e ee
e e e e e
e e e
mk nf k ne k nf c dM M m
mk km M m M M M m
k k kM m m M M M m
k k kM m m
σ σ σσσ σ σ σ
σ σ σ σ σσσ σ σ
σ σ σσσ
2 2 2
2 2 2
2
+ − + + =−
= + − + − − + + −
= − + + − − + − + −
= − − + + + +
( ) ( )2 2*22* 2 21e e
e
M m
kkM m M mσ σ
σ σ σ2
−
= − − + + + −
Next, we simplify the term
( )
( ) ( )
( )
( )
22 2 2 2 2
2 2 22 2 2 2
22 2
1
1 1 1
1 1 1
1 1
ee e
e e ee
ee
me f c dn M M m
mnm n M n M M m
nm n M n M M m
nm n M m
σσ σ σ σ σ
σ σ σσ σ σ σ
σσ σ
− + + = −
= + − − + + = −
= + − − + − + = −
= + + −
and the term
C03ed19.doc 7/8/2003 4:11 PM 10
( )2 2
2 2 2
2 22 2 2
2 2 2
2 2 2
1 1
1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
ee
ee
e
e
ce fN n nm n M N n
n m nm n n M N n
n N n n m M nm
n N n n m M nm
σ σσ σ σ
σ σσ σ σ
σ σ σ
σ σ σ
+ + = + + − + − −
= + + − +−
= + + − + − = + + − + −
.
Substituting these terms in the expression for the common MSE term, ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
1, 1, 2, 2,
1, 2,
2 2*222 2 * 2 2
1, 22 2
1
1 1
II II II II
II II
e ee n M m e n M m
II
ee n M m
a b c d e f c d e f
e f e f
kkM m M m
nm n M m
σ σσ σ σ σ
σσ σ
− −
−
′ ′+ + + + − + + + + ′ ′+ + + − =
⊗ + − − + + + ⊗ − ′=
+ + + ⊗ −
g A B C E G g g D F L g
g K H g
I P P J
g
J J
( ) ( )
( ) ( )
( )( ) ( )
1,
2 2
2 22, 2,2 2 2
2 21, 2,
1 1
2
II
e N n M N n M
II IIee N n M
II e IIn M m N n M
n M nm N n
nm
σ σ
σ σσ σ σ
σ σ
− −
−
− × −
⊗ + ⊗
′+ + − + + + ⊗ − ′+ +
g
I P P J
g gJ J
g J g
. Thus, the common term in the MSE is given by
C03ed19.doc 7/8/2003 4:11 PM 11
( )( )
( )( ) ( )
( ) ( )( )
2 22*2 2 *
1, 1, 1,
2
2n M m
e eII n M m II II II
N n M nm
M mn M m n M mn M m N n M
II II e IIN n MN n M n M m
M mk kM m m M
mM M m
σ σσσ σ
σ σ
−2
−− ×
−− −− × −2
−− × −
⊗ − ′ ′+ + ⊗ − −
⊗ ⊗ + − ′ ′ + + ⊗
P Jg P J g g g0
JI J 0 I Pg g g
0 I J( ) ( )
( ) ( )( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2 2 2
21,
1
n M m N n MII
N n MN n M n M m
n M m n M m n M m N n Me e
II NM nm II II II
N n M n M m N n M N n M
e II n M
nm n Mσ σ σ
σ
σ
− × −
−− × −
− × − − × −2
−
− × − − × −
−
= ⊗
− − + ′ ′+ + −
′ = ⊗
0g
0 I P
J Jg J g g gJ J
g I P( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
2 2*222 * 2 21, 1, 1,
22 2 2
1, 1, 2, 2,
2 22 2 2 2
2, 2,
1
1 1
1 1
e em II e II n M m II
ee II n M m II e II N n M II
eII N n M II e
kkM m M m
nm n M m
n M nm
σ σσ σ σ
σσ σ σ
σ σσ σ σ σ
−
− −
−
′+ − − + + + ⊗ −
′ ′ + + + ⊗ + ⊗ −
′ + ⊗ + − + + +
g g P J g
g J J g g I P g
g P J g ( )
( )( ) ( )
2, 2,
2 21, 2,
2
II N n M II
e II IIn M m N n M
N n
nmσ σ
−
− × −
′ ⊗
− ′+ +
g J J g
g J g
. We explore further simplifications of the term
( ) ( )2 2*222 * 2 21e e
ekk
M m M mσ σ
σ σ σ
− − + + + − . Using the fact that
*2*
*2 2 2e
mkm
σσ σ σ
=+ +
, (1.5)
2 2*
*2 2 21 e
e
km
σ σσ σ σ
+− = −
+ +, (1.6)
and 2
*2 2 e
Mσ
σ σ
= −
, (1.7)
C03ed19.doc 7/8/2003 4:11 PM 12
( ) ( ) ( )
( ) ( )
( )( ) ( )
( ) ( )
22 2 2 2 2*2 *222 * 2 2 *2 2 2*2 2 2
2 2 2*2 *22 2 2 2
*2 2 2 *2 2 2
2* *2* 2 2 2 2
*2 2 *
1
1
1
e e e ee e
e
e ee e
e e
ee e
e
k kkM m M m m M mm
km M mm m
k kkm m M mk k km
σ σ σ σ σσ σ σ σ σ σ
σ σ σ
σ σ σσ σ σ σ σσ σ σ σ σ σ
σσ σ σ σ
σ σ
+− − + + + = + + + − −+ +
+= + + + + −+ + + +
= − + + + +−
= + − +
( )
2*
2*2 2
e
ee
M mkm M m
σ
σσ σ
+ −
+ +−
. Furthermore,
( ) ( ) ( )2 2* *2
2 2 2 2 *2 *2 **2 2 2 *2 2 2
1 1ee e
e e
k m km m m m
σ σσσ σ σ σ σ σσ σ σ σ σ σ
++ = + = = − + + + +
.
Then
( ) ( ) ( )2 2 2*222 * 2 2 *2 *1 1e e e
ekk k
M m M m M mσ σ σ
σ σ σ σ
− − + + + = − + − − .
Using this expression, and (1.7), the common term in the MSE is given by
C03ed19.doc 7/8/2003 4:11 PM 13
( )( )
( )( ) ( )
( ) ( )( )
2 22*2 2 *
1, 1, 1,
2
2n M m
e eII n M m II II II
N n M nm
M mn M m n M mn M m N n M
II II e IIN n MN n M n M m
M mk kM m m M
mM M m
σ σσσ σ
σ σ
−2
−− ×
−− −− × −2
−− × −
⊗ − ′ ′+ + ⊗ − −
⊗ ⊗ + − ′ ′ + + ⊗
P Jg P J g g g0
JI J 0 I Pg g g
0 I J( ) ( )
( ) ( )( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2 2 2
21,
1
n M m N n MII
N n MN n M n M m
n M m n M m n M m N n Me e
II NM nm II II II
N n M n M m N n M N n M
e II n M
nm n Mσ σ σ
σ
σ
− × −
−− × −
− × − − × −2
−
− × − − × −
−
= ⊗
− − + ′ ′+ + −
′ = ⊗
0g
0 I P
J Jg J g g gJ J
g I P( ) ( ) ( )
( ) ( )
( ) ( )
2*2 *
1, 1, 1,
2 2 22
1, 1, 2, 2,
2 2*2 22
2, 2, 2, 2,
2 2
1
1
2
em II II n M m II
e eII n M m II e II N n M II
eII N n M II II N n M II
e
kM m
nm n M m
n nm N n
nm
σσ
σ σ σσ
σ σσ σσ
σ σ
−
− −
− −
′+ − + ⊗ −
+′ ′ + + ⊗ + ⊗ −
+′ ′ + ⊗ + + + ⊗ −
++
g g P J g
g J J g g I P g
g P J g g J J g
( ) ( )1, 2,II IIn M m N n M− × −
′
g J g
Simplification of Expressions for the Remaining Terms in the MSE for T . The remaining terms in the expression for the MSE for T are given by
C03ed19.doc 7/8/2003 4:11 PM 14
( ) ( ) ( )( )
( ) ( )
( )
( )
( )
* * * * *2
2 * * * *1,
*
1 1 1
2 1
2
m nmI e n m r r e r n I
M m mr e II n I
n M m m NM nm nmII I
N n M nm
k k k k k k km nm
k k k k km
km nm
σ − ×
− × − ×
− ×
′ ⊗ + − + − − + ⊗ +
′+ − − − ⊗
⊗ ′+ +
J Jg I P P g
Jg P g
P J Jg g0
( ) ( ) ( )( )
( ) ( )
( )
( )
( )
* * * * *2
2 * * * *1,
*
1 1 1
2 1
2 2
m nme I n m I r r e r I n I I I
M m mr e II n I
n M m m NM nm nmII I II I
N n M nm
k k k k k k km nm
k k k k km
km nm
σ − ×
− × − ×
− ×
=
′ ′ ′⊗ + − + − − + ⊗ +
′+ − − − ⊗
⊗ ′ ′+ +
J Jg I P g g P g g g
Jg P g
P J Jg g g g0
. We use a similar strategy to simplify these terms. First, using ( )1, 2,II II II′ ′ ′=g g g , we express
( )
( )( ) ( )
( )( )1, 2, 1,
n nM m m M m m
II I II II I II n IM m mN n M nm N n M nm
− × − ×
− ×− × − ×
⊗ ⊗ ′ ′ ′ ′= = ⊗
P J P Jg g g g g g P J g0 0 . Then,
combining terms,
C03ed19.doc 7/8/2003 4:11 PM 15
( ) ( ) ( )( )
( ) ( )
( )
( )
( )
* * * * *2
2 * * * *1,
*
1 1 1
2 1
2
m nmI e n m r r e r n I
M m mr e II n I
n M m m NM nm nmII I
N n M nm
k k k k k k km nm
k k k k km
km nm
σ − ×
− × − ×
− ×
′ ⊗ + − + − − + ⊗ +
′+ − − − ⊗
⊗ ′+ +
J Jg I P P g
Jg P g
P J Jg g0
( ) ( ) ( )( )
( ) ( )
( ) ( )
* * * * *2
2 * * * *1,
*1,
1 1 1
2 1
2 2
m nme I n m I r r e r I n I I I
M m mr e II n I
M m m NM nm nmII n I II I
k k k k k k km nm
k k k k km
km nm
σ − ×
− × − ×
=
′ ′ ′⊗ + − + − − + ⊗ + ′= + − − − ⊗ ′ ′+ ⊗ +
J Jg I P g g P g g g
Jg P g
J Jg P g g g
which simplifies to
( ) ( ) ( )( )
( ) ( )
( )
( )
( )
* * * * *2
2 * * * *1,
*
1 1 1
2 1
2
m nmI e n m r r e r n I
M m mr e II n I
n M m m NM nm nmII I
N n M nm
k k k k k k km nm
k k k k km
km nm
σ − ×
− × − ×
− ×
′ ⊗ + − + − − + ⊗ +
′+ − − − ⊗
⊗ ′+ +
J Jg I P P g
Jg P g
P J Jg g0
( ) ( ) ( )( )
( ) ( ) ( )
* * * * *2
2
* * * *1,
1 1 1
2 1 1 2
m nme I n m I r r e r I n I I I
M m m NM nm nmr e II n I II I
k k k k k k km nm
k k k k km nm
σ− × − ×
=
′ ′ ′⊗ + − + − − + ⊗ + =
′ ′+ − − − + ⊗ +
J Jg I P g g P g g g
J Jg P g g g
To summarize, the remaining term in the expression for the MSE for T is given by
C03ed19.doc 7/8/2003 4:11 PM 16
( )
( ) ( )( )
( ) ( ) ( )
2 * * * * *2
* * * *1,
1 1 1
2 1 1 2
e I n m I
m nmr r e r I n I I I
M m m NM nm nmr e II n I II I
k
k k k k k km nm
k k k k km nm
σ
− × − ×
′ ⊗ ′ ′+ − + − − + ⊗ + ′ ′+ − − − + ⊗ +
g I P g
J Jg P g g g
J Jg P g g g
. (1.8)
Combining expressions, the MSE for T is given by
( )( ) ( ) ( )
( ) ( )
( )
1 2 3
22 *2 *
1, 1, 1, 1,
2 2 22
1, 1, 2, 2,
2 2*2 22
2, 2,
ˆvar
1
1
ee II n M m II II n M m II
e eII n M m II e II N n M II
eII N n M II
T
kM m
nm n M m
n nm N n
ξ ξ ξ
σσ σ
σ σ σσ
σ σσ σσ
− −
− −
−
=
′ ′ ⊗ + − + ⊗ −
+′ ′ + + ⊗ + ⊗ −
+′ + ⊗ + + + −
g I P g g P J g
g J J g g I P g
g P J g ( )
( ) ( )
( )
( ) ( )( )
( )
2, 2,
2 2
1, 2,
* * * * *2
2* * * *
1,
2
1 1 1
2 1 1
II N n M II
eII IIn M m N n M
e I n m I
m nmr r e r I n I I I
r e
nm
k
k k k k k km nm
k k k k k
σ σ
σ
−
− × −
+
′ ⊗
+ ′+ ′ ⊗
′ ′+ − + − − + ⊗ +
+ ′+ − − − +
g J J g
g J g
g I P g
J Jg P g g g
g ( )
( )2
M m mII n I
NM nm nmII I
m
nm
− ×
− ×
⊗
′+
JP g
Jg g
(1.9)
while the MSE of AT is given by
C03ed19.doc 7/8/2003 4:11 PM 17
( )( ) ( ) ( )
( ) ( )
( )
1 2 3
22 *2 *
1, 1, 1, 1,
2 2 22
1, 1, 2, 2,
2 2*2 22
2, 2,
ˆvar
1
1
A
ee II n M m II II n M m II
e eII n M m II e II N n M II
eII N n M II
T
kM m
nm n M m
n nm N n
ξ ξ ξ
σσ σ
σ σ σσ
σ σσ σσ
− −
− −
−
=
′ ′ ⊗ + − + ⊗ −
+′ ′ + + ⊗ + ⊗ −
+′ + ⊗ + + + −
g I P g g P J g
g J J g g I P g
g P J g ( )
( ) ( )
[ ]
2, 2,
2 2
1, 2,
2
2
II N n M II
eII IIn M m N n M
II II
nmσ σ
σ
−
− × −
+
′ ⊗
+ ′+ ′+
g J J g
g J g
g g
.(1.10)
Special Cases Linear Combinations of PSU Averages
We first assume that Mi M
′′′ = ⊗
1g e where ie denotes an N -dimensional column
vector. If the values of ie correspond to a value of one in the thi position, and zero elsewhere, then the linear combination corresponds to a linear combination of random variables for the PSU in the thi position. We do not make this additional assumption
here, and let elements of ie be general. Then mI iI
mM m
′′ ′ = ⊗
1g e ,
1,M m
II iIM mM M m
− ′− ′ ′ = ⊗ −
1g e , and ( )2,
1II iII MM
′′ ′= ⊗g e 1 . As a result,
C03ed19.doc 7/8/2003 4:11 PM 18
( )
( )
( )
( )
( )
1, 1,
2 2 2
1, 1,
2
1, 1,
2, 2,
2, 2,
0
1
0
II n M m II
II n M m II iI n iI iI iI iI n iI
II n M m II iI n iI
II N n M II
II N n M II iII N n iII iII i
M m M m M mM M n M
M mM
−
−
−
−
− −
′ ⊗ =
− − − ′ ′ ′ ′⊗ = = −
− ′ ′⊗ =
′ ⊗ =
′ ′ ′⊗ = =
g I P g
g P J g e P e e e e J e
g J J g e J e
g I P g
g P J g e P e e e
( )
( ) ( )
2, 2,
1, 2,
1II iII N n iII
II N n M II iII N n iII
II II iI iIIn M m N n M n N n
N n
M mM
−
− −
− × − × −
′−−
′ ′⊗ =
− ′ ′=
e J e
g J J g e J e
g J g e J e
.
Using these results, the common term to the MSE simplifies to
( )
( )
( )
22*2 *
22*2 *
2 2 22
2 2*2 22 2
1
1 1
1
1
eiI iI
eiI n iI
e eiI n iI
eiII iII iII N n iII
M mkM m M
M mkn M m M
M mnm n M m M
N n n nm N n
σσ
σσ
σ σ σ
σ σσ σσ σ −
− ′− + −
− ′− − + − + − ′ + + −
+′ ′ ′ + − + + + − −
e e
e J e
e J e
e e e J e
( )2 22
iII N n iII
eiI iIIn N n
M mnm M
σ σ
−
× −
+ − ′ +
e J e
e J e
which we simplify to
( )
( ) ( )
( )
22*2 *
*2 * 2 2 22 2
2 2*2 2 22
2 2
1
1 1 1
2
eiI iI
ee eiI n iI
eiII iII iII N n iII
e
M mkM m M
k M mn n M m nm n M m M
n nm N n N n
M mnm M
σσ
σ σ σσ σ
σ σσ σ σσ
σ σ
−
− ′− + − − + − ′ + − − + + − −
+′ ′ + + + + − − −
+ − +
e e
e J e
e e e J e
iI iIIn N n× −
′ e J e
or
C03ed19.doc 7/8/2003 4:11 PM 19
( )
( ) ( )
( )
22*2 *
*2 * 2 2 2
2 2*22
2 2
1
1
2
eiI iI
eiI n iI
eiII iII iII N n iII
eiI iIIn N n
M mkM m M
k M mn nm M
n nm
M mnm M
σσ
σ σ σ
σ σσσ
σ σ
−
× −
− ′− + − − + − ′ + − +
+
′ ′ + + +
+ − ′ +
e e
e J e
e e e J e
e J e
.
We add and subtract terms to re-express this as
( ) ( )
( ) ( ) ( )
2 22 2*2 * *2 *
*2 * 2 22 22*2 *
22
11 1
11 1
e eiI iI iI n iI
eeiI n iI iI n iI
iII iII
M m M mk kM m M n M m M
kM m M mkn M m M n nm M
n
σ σσ σ
σ σ σσσ
σσ
− − ′ ′− + − − + − − − + − − ′ ′ + − + + − + −
′ + −
e e e J e
e J e e J e
e e
( )
( )
2 22 *2
2 2
22 2*2 * 2
2
1 1 11
eiII N n iII iII N n iII iII N n iII
eiI iIIn N n
eiI n iI e
n n nm
M mnm M
M m M mkM m M n M m m nm M
σ σσ σ
σ σ
σ σσ σ
− − −
× −
+′ ′ ′+ + +
+ − ′ +
− − ′= − + + + + − −
e J e e J e e J e
e J e
e P e
( )
( )
2
2 22 2 2*22
2 22*2 * 2 2
2
2
11
iI n iI
eeiII N n iII iII N n iII iI iIIn N n
eiI n iI e iI n iI
m M mn nm nm M
M m M M mkM m M nm M m M
σ σσ σ σσσ
σσ σ σ
σ
− − × −
′
+ + + − ′ ′ ′ + + + +
− − ′ ′= − + + + − −
′+
e J e
e P e e J e e J e
e P e e J e
e( )2 22 2 2
*221 ee
iII N n iII iII N n iII iI iIIn N n
m M mn m nm M
σ σσ σ σσ− − × −
+ + + − ′ ′ + + +
P e e J e e J e
. Thus, the common term in the expression for the MSE simplifies to
C03ed19.doc 7/8/2003 4:11 PM 20
( ) ( ) ( )
( ) ( )
( )
22 *2 *
1, 1, 1, 1,
2 2 22
1, 1, 2, 2,
2 2*2 22
2, 2, 2,
1
1
ee II n M m II II n M m II
e eII n M m II e II N n M II
eII N n M II II N
kM m
nm n M m
n nm N n
σσ σ
σ σ σσ
σ σσ σσ
− −
− −
− −
′ ′ ⊗ + − + ⊗ −
+′ ′ + + ⊗ + ⊗ −
+′ ′ + ⊗ + + + −
g I P g g P J g
g J J g g I P g
g P J g g J( )
( ) ( )
( )
2,
2 2
1, 2,
2 22*2 * 2 2
22 *2
2
11
1
n M II
eII IIn M m N n M
eiI n iI e iI n iI
iII N n iII
nm
M m M M mkM m M nm M m M
mn
σ σ
σσ σ σ
σ σσ σ
− × −
−
=
⊗ + ′+ − − ′ ′= − + + + − −
+′+ + +
J g
g J g
e P e e J e
e P e( )2 22 2 2 ee
iII N n iII iI iIIn N n
M mm nm M
σ σσ− × −
+ + − ′ ′ +
e J e e J e
.
Next, we simplify the remaining term. When mI iI
mM m
′′ ′ = ⊗
1g e ,
1,M m
II iIM mM M m
− ′− ′ ′ = ⊗ −
1g e , ( )2,
1II iII MM
′′ ′= ⊗g e 1 and
( )1II iI M m iII MM −
′ ′′ ′ ′= ⊗ ⊗g e 1 e 1 , then
( )
( )
2
2
1,
0
1 1
1
1
1
I n m I
mI n I iI n iI iI n iI
nmI I iI n iI
M m mII n I iI n iI
II II iI iI iII iII
m mm m M M M
mnm nm M
M mm M M
M mM M
− ×
′ ⊗ =
′ ′ ′⊗ = =
′ ′=
− ′ ′⊗ = − ′ ′ ′= +
g I P g
Jg P g e P e e P e
Jg g e J e
Jg P g e P e
g g e e e e
and
C03ed19.doc 7/8/2003 4:11 PM 21
( ) ( ) ( )
( )( )
1 1
1 1
1
NM nm nm mII I iI M m iII M iINM nm nm
miI n m iII m iIN n n
iI n iI iII iIN n n
mnm nm M M mm M m M
nm M M m
M mnM M
− ×− − ×
− ×
− ×
′ ′′ ′ ′= ⊗ ⊗ ⊗
′ ′ ′ ′= ⊗ − ⊗ ⊗
− ′ ′= +
J 1g g e 1 e 1 J e
1e J 1 e J 1 e
e J e e J e
.
The remaining terms for the MSE of T simplifies to
( )
( ) ( )( )
( ) ( )
( )
( ) ( )( )
* * * * *2
2* * * *
1,
* * * * *2
2
1 1 1
2 1 1
2
1 1 1
e I n m I
m nmr r e r I n I I I
M m mr e II n I
NM nm nmII I
r r e r
k
k k k k k km nm
k k k k km
nm
k k k k k k
M
σ
σ
− ×
− ×
′ ⊗
′ ′+ − + − − + ⊗ +
′+ − − − + ⊗
′+
− + − − +
g I P g
J Jg P g g g
Jg P g
Jg g
( )* * * *
1
2 1 1
2 2
iI n iI
iI n iI
r e iI n iI
iI n iI iII iIN n n
mM
mn M
M mk k k k kM
M mn M n − ×
′ ′+
− ′+ − − − + − ′ ′+ +
e P e
e J e
e P e
e J e e J e .
Combining like terms,
C03ed19.doc 7/8/2003 4:11 PM 22
( )
( ) ( )( )
( ) ( )
( )
( ) ( )( )
* * * * *2
2* * * *
1,
* * * * *2
2
1 1 1
2 1 1
2
1 1 1
e I n m I
m nmr r e r I n I I I
M m mr e II n I
NM nm nmII I
r r e r
k
k k k k k km nm
k k k k km
nm
k k k k k k
M
σ
σ
− ×
− ×
′ ⊗
′ ′+ − + − − + ⊗ +
′+ − − − + ⊗
′+
− + − − +
g I P g
J Jg P g g g
Jg P g
Jg g
( )* * * *2 1 1
1 2
2
iI n iI
r e
iI n iI
iII iIN n n
mM
M mk k k k kM
m M mn M M
n − ×
′ − + − − − + − ′ + + ′+
e P e
e J e
e J e
and as a result,
or
( )
( ) ( )( )
( ) ( )
( )
( ) ( )( )
* * * * *2
2* * * *
1,
* * * * *2
2
1 1 1
2 1 1
2
1 1 1
e I n m I
m nmr r e r I n I I I
M m mr e II n I
NM nm nmII I
r r e r
k
k k k k k km nm
k k k k km
nm
k k k k k k
M
σ
σ
− ×
− ×
′ ⊗
′ ′+ − + − − + ⊗ +
′+ − − − + ⊗
′+
− + − − +
=
g I P g
J Jg P g g g
Jg P g
Jg g
( )* * * *2 1 1
1 22
iI n iI
r e
iI n iI iII iIN n n
mM
M mk k k k kM
mn M n − ×
′ − + − − − + ′ ′ + − +
e P e
e J e e J e
. It may be possible to simplify this further. We do not do so.
C03ed19.doc 7/8/2003 4:11 PM 23
To summarize, the MSE for T is given by
( )
( )
( )
1 2 3
22*2 *
22 2
2 2 22 *2
2 2
ˆvar
1
1
1
2
eiI n iI
e iI n iI
eiII N n iII iII N n iII
eiI iIIn N n
T
M mkM m M
M M mnm M m M
mn m
M mnm M
ξ ξ ξ
σσ
σ σ
σ σ σσ σ
σ σ
− −
× −
=
− ′− + −
− ′+ + − + +
′ ′+ + +
+ − ′ +
e P e
e J e
e P e e J e
e J e
( ) ( )( )
( )
* * * * *2
2* * * *
1 1 1
2 1 1
1 22
r r e r
iI n iI
r e
iI n iI iII iIN n n
mk k k k k kM
M mk k k k kMM
mn M n
σ
− ×
+
− + − − + ′ − + − − − ++ ′ ′ + − +
e P e
e J e e J e
. (1.11)
The MSE for AT is given by
( )
( )
( )
1 2 3
22*2 *
22 2
2 2 22 *2
2 2
ˆvar
1
1
1
2
A
eiI n iI
e iI n iI
eiII N n iII iII N n iII
eiI iIIn N n
T
M mkM m M
M M mnm M m M
mn m
M mnm M
ξ ξ ξ
σσ
σ σ
σ σ σσ σ
σ σ
− −
× −
=
− ′− + −
− ′+ + − + +
′ ′+ + +
+ − ′ +
e P e
e J e
e P e e J e
e J e
2
iI iI iII iIIM m
M Mσ
+
− ′ ′+ + e e e e
. (1.12)
Simplifications when m M<<
C03ed19.doc 7/8/2003 4:11 PM 24
We consider the special case where m M<< so that 1M mM−
≅ and
1 1 0M M m
≅ ≅−
. Then the MSE for T simplifies to
( )
( )
1 2 3
2 2*2 *
2 2 22 *2
2 2
1
1ˆvar
2
eiI n iI iI n iI
eiII N n iII iII N n iII
eiI iIIn N n
knm
mT
n m
nm
ξ ξ ξ
σ σσ
σ σ σσ σ
σ σ
− −
× −
+ ′ ′− + + + ′ ′= + + +
+ ′+
e P e e J e
e P e e J e
e J e
while the MSE for AT is identical and equal to
( )
( )
( )1 2 3
*2 * 2 2
2 2 22 *2
2 2
11
1ˆvar
2
iI n iI e iI n iI
eA iII N n iII iII N n iII
eiI iIIn N n
knmm
Tn m
nm
ξ ξ ξ
σ σ σ
σ σ σσ σ
σ σ
− −
× −
′ ′ − + + + + +
′ ′= + + +
+ ′ +
e P e e J e
e P e e J e
e J e
Notice that unless the sum of the coefficients of iII′e are bounded, the common term for the MSE will be infinite. Linear Combinations corresponding to a single PSU Average
We assume that Mi M
′′′ = ⊗
1g e where ie denotes an N -dimensional column
vector where the values of ie correspond to a value of one in the thi position, and zero elsewhere, then the linear combination corresponds to a linear combination of random variables for the PSU in the thi position. This additional assumption implies that when
i n≤ , 11iI n iI n ′ = −
e P e , 1iI n iI′ =e J e , 0iII N n iII−′ =e P e , 0iII N n iII−′ =e J e , 0iI iIIn N n× −′ =e J e ,
and when
i n> 0iI n iI′ =e P e , 0iI n iI′ =e J e , 11iII N n iII N n− ′ = − −
e P e , 1iII N n iII−′ =e J e , 0iI iIIn N n× −′ =e J e .
Then, when i n≤
C03ed19.doc 7/8/2003 4:11 PM 25
( )
( )
( ) ( )( )
( )
1 2 3
2 2*2 * 2 2
* * * * *22
* * * *
ˆvar
1 11 1
1 1 11 11 2
2 1 1
ee
r r e r
r e
T
M m MkM M m n nm M m
mk k k k k kM m
M n n MM mk k k k kM
ξ ξ ξ
σσ σ σ
σ
=
− − + − + + + − −
− + − − + + − + − − + − − − +
(1.13)
and when i n> ,
( )1 2 3
2 2 2*22 1ˆvar 1 em
TN n n nmξ ξ ξ
σ σ σσσ+ + = − + + −
.
The MSE for AT can be developed in a similar manner, resulting in when i n≤
( )( )
1 2 3
2*2 *
2
2 2
2
11 1ˆvar
1
e
A
e
kM m nM mT
M Mnm M m
M mM M
ξ ξ ξ
σσ
σ σ
σ
− + − − − = + + + − − +
(1.14)
and when i n> ,
( )1 2 3
2 2 2*2 22 1ˆvar 1 e
Am
TN n n nm Mξ ξ ξ
σ σ σσ σσ+ + = − + + + −
.(1.15)
More General Linear Combinations of Random Variables Similar developments for more general linear combinations of random variables
are given in c02ed27.doc. These developments are based on defining ( )i M′′ = ⊗g e I a
where ( )( )i ie=e denotes an N -dimensional column vector, and where
( )1 2 N′′ ′ ′=a a a a and ( )( )i ija=a is an 1M × column vector of constants. We can
represent g equivalently as
1
N
N i iie
=′ ′ ′= ⊕g 1 a . (1.16)
C03ed19.doc 7/8/2003 4:11 PM 26
Post multiplying ′g by ( )I II′ ′ ′=K K K will result in ( )I II′ ′ ′ ′=g K g g , where
( ) ( )1, 2, 1, 2,II II II II II II′ ′ ′ ′ ′ ′ ′ ′ ′= = =g K g g K g K g g . We partition ( )i iI iII′′ ′=a a a , where iI′a
is of dimension 1 m× , and iII′a is of dimension ( )1 M m× − . Then
( )
1
n
i iIiI I N
N n nm
e=
− ×
′⊕ ′ ′ ′ ′= =
ag K g 1
0,(1.17)
and 1
1
n
i iII NiII N
i ii n
ee
=
= +
′⊕ ′ ′= ′⊕
0ag 1
a0 where 1
1,
n
i iIIiII N
e=
′⊕ ′ ′=
ag 1
0and 2,
1
NII N
i ii ne
= +
′ ′= ′⊕
0g 1
a.
The predictor of T (given by (1.1) is once again given by (1.2), with the exception that ′g is given by (1.16). As a result, the additional expressions for the predictor and MSE
are identical, with the exception of the different definition of the term for g .
