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C06ed16.doc 5/1/2006 3:32 PM 1
Summary of Expanded Model and Unbiased Constraint with Unequal Size Clusters Ed Stanek
We summarize the development of the unbiased constraint given in c06ed15.doc in a
setting where clusters differ in size. The mixed model may be expressed as
( )1 1 1
1 1s s
N N N
N N M Ms i sN Nµ
= = =
⎡ ⎤⎛ ⎞ ⎡ ⎤⎛ ⎞ ⎛ ⎞= + ⊗ ⊕ + ⊕⊕ + +⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦⎣ ⎦Y 1 1 1 β 1 M B E
where ( )1 2 N′′ ′ ′=E E E E . An element of Y is given by ( )
1
sMs
is sj is jt stt
U Y U U y=
= ∑ . Let us
define a weight,
1 1
s
sjsj MN
sjs j
ww
w= =
=
∑∑ for 1,...,s N= , 1,..., sj M= such that
1 1
1sMN
sjs j
w= =
=∑∑ .
For example, suppose that 1sj
s
wM
= for 1,...,s N= , 1,..., sj M= , where 1 1
sMN
sjs j
w N= =
=∑∑ .
Then 1sj
s
wNM
= or equivalently, since ss
MrM
= . Then 1sj
s
wr
= where NM= .
As a second example, when the sample SSUs are divided by the cluster sample size, and
the remainder SSUs are divided by the remaining number of SSUs, 1sj
s
wm
= for 1,..., sj m= and
( )1
sjs s
wM m
=−
for 1,...,s sj m M= + . In this setting,
( ) ( )1 1 1 1 1 1 1 1 1
1 1 1 1 2 2s s s s s
s s
M m M m MN N N N
sjs j s j j m s j j m ss s s s s s
w Nm M m m M m= = = = = + = = = + =
⎛ ⎞ ⎛ ⎞= + = + = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠
∑∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ . Then
12sj
s
wNm
= for 1,..., sj m= and ( )
12sj
s s
wN M m
=−
for 1,...,s sj m M= + . Recall that we have
defined ssI
mrm
= and s ssII
M mrM m
−=
−. Using these expressions, 1
2sjsI
wNmr
= for 1,..., sj m= and
( )1
2sjsII
wN M m r
=−
for 1,...,s sj m M= + .
Finally, when 1sjw = , 1 1
sMN
sjs j
w= =
=∑∑ , so that 1sjw = . Thus, we can consider each of
these situations a special case of a weighted set of values of is sjU Y where ( )
1
sMs
sj jt stt
Y U y=
= ∑ .
C06ed16.doc 5/1/2006 3:32 PM 2
Recall that elements of Y are given by ( )
1
sMs
is sj is jt stt
U Y U U y=
= ∑ . Let us define
( )( ) ( )1 2w wi w w wN′′ ′ ′= =Y Y Y Y Y where ( )( ) ( )1 1 2 2wi is ws i w i w iN wNU U U U ′′ ′ ′= =Y Y Y Y Y
and ( )( ) ( )1 2 sis ws wisj wis wis wisMU Y Y Y Y ′= =Y with elements given by wisj is sj sjY U w Y= . A
mixed model is expressed as
( ) ( ) ( ) ( )1 2 2 1 2 2wisj wisj wisj wisj wisj wisjY E Y E Y E Y Y E Yξ ξ ξ ξ ξ ξ⎡ ⎤ ⎡ ⎤= + − + −⎣ ⎦ ⎣ ⎦ .
Taking the expected value of SSUs, ( )2 wisj is sj sE Y U wξ µ= , while ( )1 2
1wisj sj sE Y w
Nξ ξ µ= . As a
result,
( )1 1wisj sj s is sj s is sj sj sY w U w U w Y
N Nµ µ µ⎡ ⎤ ⎡ ⎤= + − + −⎢ ⎥ ⎣ ⎦⎣ ⎦
.
Let us consider these expressions for different values of sjw . When 1sjw = ,
( )2
swisj isE Y Uξ
µ= , while ( )1 2
swisjE Y
Nξ ξµ
= . The model is given by
( )1 1 1swisj s is is sj sY U U Y
N Nµ
µ µ⎡ ⎤ ⎡ ⎤= + − + −⎢ ⎥ ⎣ ⎦⎣ ⎦.
When 1sj
s
wr
= , ( )2
swisj is
s
E Y Urξµ
= , while ( )1 2
1 swisj
s
E YN rξ ξµ
= . The model is given by
1 1 1 1 sjs s swisj is is
s s s s
YY U U
N r N r r rµ µ µ⎡ ⎤⎛ ⎞⎡ ⎤= + − + −⎢ ⎥⎜ ⎟⎢ ⎥ ⎜ ⎟⎣ ⎦ ⎢ ⎥⎝ ⎠⎣ ⎦
.
When 12sj
sI
wNmr
= for 1,..., sj m= and ( )1
2sjsII
wN M m r
=−
for 1,...,s sj m M= + ,
then when 1,..., sj m= , ( )2
12
swisj is
sI
E Y UNm rξ
µ= , while ( )1 2
1
1 12
Ns
wisjs sI
E YNm N rξ ξ
µ
=
⎛ ⎞= ⎜ ⎟
⎝ ⎠∑ while
when 1,...,s sj m M= + , ( ) ( )2
12
swisj is
sII
E Y UrN M mξµ
=−
, while
( ) ( )1 21
1 12
Ns
wisjs sII
E YN rN M mξ ξ
µ
=
⎛ ⎞= ⎜ ⎟
− ⎝ ⎠∑ .
The model is given by
C06ed16.doc 5/1/2006 3:32 PM 3
1 1 1 1 12 2 2
sjs s swisj is is
sI sI sI sI
YY U U
N Nm r Nm N r Nm r rµ µ µ⎡ ⎤⎛ ⎞⎡ ⎤= + − + −⎢ ⎥⎜ ⎟⎢ ⎥ ⎜ ⎟⎣ ⎦ ⎢ ⎥⎝ ⎠⎣ ⎦
when 1,..., sj m= , and
( ) ( ) ( )1 1 1 1 1
2 2 2sjs s s
wisj is issII sII sII sII
YY U U
N r N r r rN M m N M m N M mµ µ µ⎡ ⎤⎛ ⎞⎡ ⎤= + − + −⎢ ⎥⎜ ⎟⎢ ⎥ ⎜ ⎟− − −⎣ ⎦ ⎢ ⎥⎝ ⎠⎣ ⎦
when
1,...,s sj m M= + .
Consider the general mixed model given by
( )1 1wisj sj s is sj s is sj sj sY w U w U w Y
N Nµ µ µ⎡ ⎤ ⎡ ⎤= + − + −⎢ ⎥ ⎣ ⎦⎣ ⎦
.
Now
( )( )
( )
1 11 1
2 2 2 21 1
s ss s
s s ss s
s s s s sis ws s is s is
sM sMsM sM s
w Yw ww w w Y
U U UN N
w w w Y
µ
µµ µ
µ
⎛ ⎞−⎛ ⎞ ⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎡ ⎤⎜ ⎟ ⎜ ⎟ ⎜ ⎟= + − +⎢ ⎥⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎜ ⎟−⎝ ⎠
Y . Let us define
( )( ) ( )1 2 ss sj s s sMw w w w ′= =w . Then
( )( )
( )
1 1
2 21 1
s s
s s s
s s sis ws s s is s s is
sM sM s
w Y
w YU U U
N N
w Y
µ
µµ µ
µ
⎛ ⎞−⎜ ⎟⎜ ⎟−⎡ ⎤ ⎜ ⎟= + − +⎢ ⎥⎣ ⎦ ⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠
Y w w , and
C06ed16.doc 5/1/2006 3:32 PM 4
1 1
2
11 11
12 121 1
1 1
211 1
2222 2
2
1
2
1 1
1
1
N
i
M M
i w
i wwi
MiN wN
N
NN
NM
w ww w
UN N
w w
wU wU
Nw
U
ww
Nw
µ
µ
µ
⎛ ⎞⎛ ⎞ ⎛⎜ ⎟⎜ ⎟
⎡ ⎤⎜ ⎟⎜ ⎟ −⎢ ⎥⎜ ⎟⎜ ⎟ ⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝⎜ ⎟
⎛ ⎞⎜ ⎟⎛ ⎞ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟= = +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎜ ⎟⎝ ⎠ ⎜ ⎟
⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
YY
Y
Y
( )( )
( )1 1
2
11 11 1
12 12 11
1
1 1 1
21 2121
222 2 2
2
1
2
1
1
N
i
M M
i i
M
N
NiN N
NM
w Y
w YU
w Y
w Yww
U UN
w
ww
UN
w
µ
µ
µµ
µ
µ
⎛ ⎞−⎜ ⎟
⎛ ⎞ ⎜ ⎟⎞ −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟
⎜ ⎟−⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎜ ⎟⎠⎜ ⎟
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤ ⎜ ⎟⎜ ⎟−⎢ ⎥ ⎜ ⎟ +⎜ ⎟⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟
⎜ ⎟⎜ ⎟
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟
⎡ ⎤ ⎜ ⎟⎜ ⎟−⎢ ⎥ ⎜ ⎟⎜ ⎟⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
( )( )
( )
( )( )
( )
2 2
2
22 22 2
2 2 2
1 1
2 2
N N
M M
N N N
N N NiN
NM NM N
w Y
w Y
w Y
w YU
w Y
µ
µ
µ
µ
µ
µ
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
⎜ ⎟−⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎜ ⎟⎝ ⎠⎝ ⎠
or
1 1 11 1
1 1
2 2 22 22 2
11
11
1 1
i
i w
ii wwi wi
iN wN
N N iN N N
UNNU
UUNN
UUN N
µµ
µµ
µ µ
⎛ ⎞⎡ ⎤⎛ ⎞ −⎜ ⎟⎢ ⎥⎜ ⎟ ⎣ ⎦⎜ ⎟⎛ ⎞ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎡ ⎤⎜ ⎟ −⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟= = + +⎣ ⎦⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎡ ⎤⎜ ⎟⎜ ⎟ −⎢ ⎥⎜ ⎟⎝ ⎠ ⎣ ⎦⎝ ⎠
wwY
wwYY E
Yw w
.
where
C06ed16.doc 5/1/2006 3:32 PM 5
( )( )
( )( )( )
( )
( )( )
( )
1 1
2 2
11 11 1
12 12 11
1 1 1
21 21 2
22 22 22
2 2 2
1 1
2 2
N N
i
M M
iwi
M M
N N N
N N NiN
NM NM N
w Y
w YU
w Y
w Y
w YU
w Y
w Y
w YU
w Y
µ
µ
µ
µ
µ
µ
µ
µ
µ
⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
⎜ ⎟−⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎜ ⎜ ⎟
= ⎜ ⎜ ⎟⎜ ⎜ ⎟
⎜ ⎟−⎜ ⎝ ⎠⎜⎜⎜ ⎛ ⎞−⎜ ⎜ ⎟⎜ ⎜ ⎟−⎜ ⎜ ⎟⎜ ⎜ ⎟⎜ ⎜ ⎟⎜ ⎜ ⎟−⎜ ⎝ ⎠⎝ ⎠
E⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟ .
Let us express
1 1
2 2
1
1
1
1
NN
s ss
N N
N
NN
N
µ
µµ
µ
=
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟
⎛ ⎞⎜ ⎟ = ⊕⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
w
w 1w
w
and
( )( )
1 1 1 1 1 1
2 2 2 2 2 2
1
1 1
1 1
1 1
i i
Ni is s i is
iN N N N N iN
U UN N
U UN N E
U UN N
µ µ
µ µµ
µ µ
=
⎛ ⎞ ⎛ ⎞⎡ ⎤ ⎡ ⎤− −⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎡ ⎤ ⎡ ⎤− −⎜ ⎟ ⎜ ⎟ ⎛ ⎞⎢ ⎥ ⎢ ⎥= = ⊕ −⎣ ⎦ ⎣ ⎦ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎜ ⎟− −⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠
w w
w ww U U
w w
so that
C06ed16.doc 5/1/2006 3:32 PM 6
( )( )1 1
2 2
1 1
i w
N Ni w N
wi s s s s i i wis s
iN wN
UU
EN
U
µ µ= =
⎛ ⎞⎜ ⎟
⎛ ⎞ ⎛ ⎞⎜ ⎟= = ⊕ + ⊕ − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎜ ⎟⎜ ⎟⎝ ⎠
YY 1
Y w w U U E
Y
.
Then, since ( )( ) ( )1 2w wi w w wN′′ ′ ′= =Y Y Y Y Y , and setting
( )( ) ( )1 2w wi w w wN′′ ′ ′= =E E E E E ,
( )( )
( )( )
( )( )
1
1
1
1 11 1
2 211
11
1
N NN
s s s ss s
NNN
s ss s ssw w
NNN
s s N Ns s ss
N
N s ss
EN
EN
EN
ξ
ξ
ξ
µ µ
µµ
µµ
µ
= =
==
==
=
⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⊕ ⊕ −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎛ ⎞⎛ ⎞ ⊕ −⊕ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠= + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⊕ −⊕ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠⎡ ⎛ ⎞= ⊗ ⊕⎜ ⎟⎢ ⎝ ⎠⎣
1w w U U
1w U Uw
Y E
1 w U Uw
1 w ( ) ( )( )11
NN
N s s wsvec E vec
N ξµ=
⎤ ⎡ ⎤⎛ ⎞ ⎡ ⎤+ ⊗ ⊕ − +⎜ ⎟⎥ ⎢ ⎥ ⎣ ⎦⎝ ⎠⎦ ⎣ ⎦
1I w U U E
( ) ( )( )11 1
N NN
w N s s N s s ws svec E vec
N ξµ µ= =
⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤= ⊗ ⊕ + ⊗ ⊕ − +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦
1Y 1 w I w U U E .
In this model, ( )1 2
1 1
2 21w N
N N
ENξ ξ
µµ
µ
⎛ ⎞⎜ ⎟
⎡ ⎤ ⎜ ⎟= ⊗⎢ ⎥ ⎜ ⎟⎣ ⎦⎜ ⎟⎜ ⎟⎝ ⎠
ww
Y 1 I
w
. The variance of the random effects is
( ) ( )( ) ( )1 2 1 1 21 1 1
1 1
var var
11
N N N
N s s N s s N s ss s s
N N
N s s N s ss s
vec E vec vec
N
ξ ξ ξ ξ ξµ µ µ
µ µ
= = =
= =
⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎡ ⎤ ′⊗ ⊕ − = ⊗ ⊕ ⊗ ⊕⎡ ⎤⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦⎡ ⎤⎛ ⎞ ⎛ ⎞′= ⊗ ⊕ ⊕⎜ ⎟ ⎜ ⎟⎢ ⎥− ⎝ ⎠ ⎝ ⎠⎣ ⎦
I w U U I w U I w
P w P w
C06ed16.doc 5/1/2006 3:32 PM 7
. To evaluate ( ) ( ) ( ) ( )1 2 1 2 1 1 2 1 1 2 1| | |var var var varw w w wE E Eξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= + =⎣ ⎦ ⎣ ⎦ ⎣ ⎦E E E E , notice that
( )
( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )
2 1 2 1 2 1
2 1 2 1 2 1
2 1
2 1 2 1 2 1
| 1 | 1 2 | 1
| 2 1 | 2 | 2|
| 1 | 2 |
var cov , cov ,
cov , var cov ,var
cov , cov , var
w w w w wN
w w w w wNw
wN w wN w wN
ξ ξ ξ ξ ξ ξ
ξ ξ ξ ξ ξ ξξ ξ
ξ ξ ξ ξ ξ ξ
⎛ ⎞′ ′⎜ ⎟⎜ ⎟′ ′⎜ ⎟=⎜ ⎟⎜ ⎟⎜ ⎟′ ′⎝ ⎠
E E E E E
E E E E EE
E E E E E
.
Now
( )
( )( )
( )( )( )
( )
( )( )
( )
1 1
2 1 2 1
2 2
11 11 1
12 12 11
1 1 1
21 21 2
22 22 22
| |
2 2 2
1 1
2 2
var var
N N
i
M M
iwi
M M
N N N
N N NiN
NM NM N
w Y
w YU
w Y
w Y
w YU
w Y
w Y
w YU
w Y
ξ ξ ξ ξ
µ
µ
µ
µ
µ
µ
µ
µ
µ
⎛ ⎛ ⎞−⎜ ⎜ ⎟⎜ ⎜ ⎟−⎜ ⎜ ⎟⎜ ⎜ ⎟⎜ ⎜ ⎟
⎜ ⎟−⎜ ⎝ ⎠⎜⎜ ⎛ ⎞−⎜ ⎜ ⎟⎜ ⎜ ⎟−⎜ ⎜ ⎟
= ⎜ ⎜ ⎟⎜ ⎜ ⎟
⎜ ⎟−⎜ ⎝ ⎠⎜⎜⎜ ⎛ ⎞−⎜ ⎜ ⎟⎜ ⎜ ⎟−⎜ ⎜ ⎟
⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠⎝
E
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
⎠
where
C06ed16.doc 5/1/2006 3:32 PM 8
( )( )
( )
( )( )
( )( )( )
2 1 2 1
2 1
1 1 1
2 2 2| | 1
1
2|1
var var
var
s
s s s
s
s s s s s
Ms s s s s
is is sjj
sM sM s sM s
s s
Ms s
is sjj
w Y Y
w Y YU U w
w Y Y
Y
YU w
ξ ξ ξ ξ
ξ ξ
µ µ
µ µ
µ µ
µ
µ
=
=
⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟= ⊕⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
−
−⎛ ⎞= ⊕⎜ ⎟⎝ ⎠
( )1
s
s
M
sj isj
sM s
w U
Y µ
=
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎛ ⎞⎜ ⎟⎜ ⎟ ⊕⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠
.
Since ( ) ( )
1
sMs s
sj jt st j st
Y U y=
′= =∑ U y ,
( )( )
( )
( )
( )
( )
( ) ( )( )2
1 1
2 2 1s
ss
ss s
ss s s s
s M s ss
sMsM s
Y
YE
M
Y
ξ
µ
µ
µ
⎛ ⎞⎛ ⎞− ⎛ ⎞′⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ′− ⎜ ⎟ ⎡ ⎤⎜ ⎟⎜ ⎟ = − = −⎜ ⎟ ⎣ ⎦⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟′⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠⎝ ⎠
U
Uy J y U U y
U
.
Then
( )( )
( )
( ) ( )( )( )2 1 2 1 2
1 1
2 2| |1 1
var vars s
s s
s s s
M Ms s s s s
is is sj s s sj isj j
sM sM s
w Y
w YU U w E w U
w Y
ξ ξ ξ ξ ξ
µ
µ
µ
= =
⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟− ⎛ ⎞ ⎛ ⎞⎡ ⎤⎜ ⎟ ′⎜ ⎟ = ⊕ − ⊕⎜ ⎟ ⎜ ⎟⎣ ⎦⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎜ ⎟⎜ ⎟
⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠
y U U y . Now
( )( ) ( )( )( ) ( )( )( )( )
( )( )( )
2 1 2 1
2 1
| |
|
2
var var
var v
11
11
s s
s s s s
s s
s
s ss s
sM s M s
M s M M M ss
M s M ss
s M
ec
M
M
ξ ξ ξ ξ
ξ ξ
σ
′ ′=
′ ′= ⊗ ⊗
′= ⊗ ⊗ ⊗−
⎡ ⎤′= ⊗ ⎢ ⎥−⎣ ⎦
=
U y y U
I y U I y
I y P P I y
P y P y
P
.
C06ed16.doc 5/1/2006 3:32 PM 9
As a result,
( )( )
( )2 1
1 1
2 2 2| 1 1
vars s
s
s s
s s s
M Ms s s
is is s sj M sjj j
sM sM s
w Y
w YU U w w
w Y
ξ ξ
µ
µσ
µ
= =
⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟− ⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ = ⊕ ⊕⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎜ ⎟⎜ ⎟
⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠
P . Using this result, and
the fact that for *s s≠ , the covariance is zero,
( )
( )( )
( )( )( )
( )
( )( )
( )
1 1
2 1 2 1
2 2
11 11 1
12 12 11
1 1 1
21 21 2
22 22 22
| |
2 2 2
1 1
2 2
var var
N N
i
M M
iwi
M M
N N N
N N NiN
NM NM N
w Y
w YU
w Y
w Y
w YU
w Y
w Y
w YU
w Y
ξ ξ ξ ξ
µ
µ
µ
µ
µ
µ
µ
µ
µ
⎛ ⎛ ⎞−⎜ ⎜ ⎟⎜ ⎜ ⎟−⎜ ⎜ ⎟⎜ ⎜ ⎟⎜ ⎜ ⎟
⎜ ⎟−⎜ ⎝ ⎠⎜⎜ ⎛ ⎞−⎜ ⎜ ⎟⎜ ⎜ ⎟−⎜ ⎜ ⎟
= ⎜ ⎜ ⎟⎜ ⎜ ⎟
⎜ ⎟−⎜ ⎝ ⎠⎜⎜⎜ ⎛ ⎞−⎜ ⎜ ⎟⎜ ⎜ ⎟−⎜ ⎜ ⎟
⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠⎝
E 2
1 1 1
s s
s
M MN
is s sj M sjs j jU w wσ
= = =
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟ ⎡ ⎤⎛ ⎞ ⎛ ⎞= ⊕ ⊕ ⊕⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥
⎝ ⎠ ⎝ ⎠⎟ ⎣ ⎦⎟⎟⎟⎟⎟⎟⎟
⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
⎠
P
.
Thus, ( )2 1
2| 1 1 1
vars s
s
M MN
wi is s sj M sjs j jU w wξ ξ σ
= = =
⎡ ⎤⎛ ⎞ ⎛ ⎞= ⊕ ⊕ ⊕⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
E P .
Next, let us consider the expression for ( )2 1| *cov ,wi wiξ ξ ′E E where *i i≠ . Since
( )( )
( )
( ) ( )( )2
1
2
s
s s
s s s ss
sM s
Y
YE
Y
ξ
µ
µ
µ
⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟− ⎡ ⎤⎜ ⎟⎜ ⎟ = −⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠
U U y and
( )( )
( )
( ) ( )( )2
1 1
2 2
1
s
s s
s s s
Ms s s s s
sj sj
sM sM s
w Y
w Yw E
w Y
ξ
µ
µ
µ
=
⎛ ⎞−⎜ ⎟⎜ ⎟− ⎛ ⎞ ⎡ ⎤⎜ ⎟ = ⊕ −⎜ ⎟ ⎣ ⎦⎜ ⎟ ⎝ ⎠⎜ ⎟⎜ ⎟−⎝ ⎠
U U y
C06ed16.doc 5/1/2006 3:32 PM 10
( )( )
( )( )( )
( )
( )( )
( )
1 1
2 2
11 11 1
12 12 11
1 1 1
21 21 2
22 22 22
2 2 2
1 1
2 2
N N
i
M M
iwi
M M
N N N
N N NiN
NM NM N
w Y
w YU
w Y
w Y
w YU
w Y
w Y
w YU
w Y
µ
µ
µ
µ
µ
µ
µ
µ
µ
⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
⎜ ⎟−⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎜ ⎜ ⎟
= ⎜ ⎜ ⎟⎜ ⎜ ⎟
⎜ ⎟−⎜ ⎝ ⎠⎜⎜⎜ ⎛ ⎞−⎜ ⎜ ⎟⎜ ⎜ ⎟−⎜ ⎜ ⎟⎜ ⎜ ⎟⎜ ⎜ ⎟⎜ ⎜ ⎟−⎜ ⎝ ⎠⎝ ⎠
E
( ) ( )( )( ) ( )( )
( ) ( )( )
( ) ( )( )( ) ( )( )
( ) ( )( )
1
2
2
2
2
2
2
2
1 11 1 11
2 22 2 21
1
1 11
2 22
1 1
N
s
M
i jj
M
i jj
MN N
iN Nj Nj
MN
is sjs j
N N
U w E
U w E
U w E
E
EU w
E
ξ
ξ
ξ
ξ
ξ
ξ
=
=
=
= =
⎛ ⎞⎛ ⎞ ⎡ ⎤⊕ −⎜ ⎟⎜ ⎟ ⎣ ⎦⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞ ⎡ ⎤⎟ ⊕ −⎜ ⎟⎜ ⎟ ⎣ ⎦=⎟ ⎝ ⎠⎜ ⎟
⎟ ⎜ ⎟⎟ ⎜ ⎟⎟ ⎛ ⎞⎜ ⎟⎡ ⎤⊕ −⎟ ⎜ ⎟⎜ ⎟⎣ ⎦⎝ ⎠⎟ ⎝ ⎠⎟⎟⎟⎟⎟⎟⎟
⎡ ⎤−⎣ ⎦⎡ ⎤−⎛ ⎞ ⎣ ⎦= ⊕ ⊕⎜ ⎟
⎝ ⎠
⎡ ⎤−⎣ ⎦
U U y
U U y
U U y
U U y
U U y
U U yN
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
.
As a result
( ) ( )( )( ) ( )( )
( ) ( )( )
2
2
2
1 11
2 22
1 1
sMN
wi is sjs j
N NN
E
EU w
E
ξ
ξ
ξ
= =
⎛ ⎞⎡ ⎤−⎣ ⎦⎜ ⎟⎜ ⎟⎡ ⎤−⎛ ⎞⎜ ⎟⎣ ⎦= ⊕ ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎡ ⎤−⎜ ⎟⎣ ⎦⎝ ⎠
U U y
U U yE
U U y
.
C06ed16.doc 5/1/2006 3:32 PM 11
Now
( ) ( )( )( ) ( )( )
( ) ( )( )
1
2
2
2
2
1 11 1 11
2 22 2 21
1
N
M
i jj
M
i jjwi
MN N
iN Nj Nj
U w E
U w E
U w E
ξ
ξ
ξ
=
=
=
⎛ ⎞⎛ ⎞ ⎡ ⎤⊕ −⎜ ⎟⎜ ⎟ ⎣ ⎦⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞ ⎡ ⎤⊕ −⎜ ⎟⎜ ⎟ ⎣ ⎦= ⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟
⎛ ⎞⎜ ⎟⎡ ⎤⊕ −⎜ ⎟⎜ ⎟⎣ ⎦⎝ ⎠⎝ ⎠
U U y
U U yE
U U y
. Let us express the vector of random
variables given by ( ) ( )( )21
sMs s
sj sjw Eξ=
⎛ ⎞ ⎡ ⎤⊕ −⎜ ⎟ ⎣ ⎦⎝ ⎠U U y as sQ . Then ( )2 1| *cov ,wi wiξ ξ ′E E where *i i≠ is
given by
( ) ( )
( ) ( ) ( )( )
2 1 2 1
2 1 2 1 2 1
2 1 2 1
1 1
2 2| * | 1 *1 2 *2 *
1 *1 | 1 1 1 *2 | 1 2 1 * | 1
2 *1 | 2 1 2 *2 | 2
cov , cov ,
cov , cov , cov ,
cov , cov ,
i
iwi wi i i N i N
iN N
i i i i i i N N
i i i i
UU
U U U
U
U U U U U U
U U U U
ξ ξ ξ ξ
ξ ξ ξ ξ ξ ξ
ξ ξ ξ ξ
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟′ ′ ′ ′= ⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
′ ′ ′
′ ′=
E E Q Q Q
Q
Q Q Q Q Q Q
Q Q Q( ) ( )
( ) ( ) ( )
2 1
2 1 2 1 2 1
2 2 * | 2
*1 | 1 *2 | 2 * |
cov ,
cov , cov , cov ,
i i N N
iN i N iN i N iN i N N N
U U
U U U U U U
ξ ξ
ξ ξ ξ ξ ξ ξ
⎛ ⎞⎜ ⎟
′⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟′ ′ ′⎝ ⎠
Q Q Q
Q Q Q Q Q Q
. Notice that ( )2 ss MEξ =Q 0 . As a result, ( ) ( )
2 1 2 1| * | *cov ,s s s sEξ ξ ξ ξ′ ′=Q Q Q Q . Now
( ) ( )( ) ( ) ( )( )2 2
* ** * *1 1
s sM Ms s s s
s s sj s s s jj jw E E wξ ξ= =
⎛ ⎞ ⎛ ⎞⎡ ⎤ ⎡ ⎤′ ′ ′ ′= ⊕ − − ⊕⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠Q Q U U y y U U . If *s s≠ , then the
random variables ( )sU and ( )*sU are independent, and hence ( )2 1
*| *cov ,
s ss s M Mξ ξ ×
′ =Q Q 0 . If
*s s= , we express ( ) ( )2 1 2 1| * |cov , vars s sξ ξ ξ ξ′ =Q Q Q . Thus,
( )
( )
( )
( )
2 11 2 1
2 12 1 2
2 1
2 11 2
1 *1 | 1
2 *2 | 2
| *
* |
var
varcov ,
var
N
N
N N
i i M M M M
i iM M M Mwi wi
iN i N NM M M M
U U
U U
U U
ξ ξ
ξ ξ
ξ ξ
ξ ξ
× ×
× ×
× ×
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟′ =⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
Q 0 0
0 Q 0E E
0 0 Q
or
C06ed16.doc 5/1/2006 3:32 PM 12
( ) ( )2 1 2 1| * * |1
cov , varN
wi wi is i s ssU Uξ ξ ξ ξ=
′ = ⊕E E Q .
Using these expressions, since
( )
( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )
2 1 2 1 2 1
2 1 2 1 2 1
2 1
2 1 2 1 2 1
| 1 | 1 2 | 1
| 2 1 | 2 | 2|
| 1 | 2 |
var cov , cov ,
cov , var cov ,var
cov , cov , var
w w w w wN
w w w w wNw
wN w wN w wN
ξ ξ ξ ξ ξ ξ
ξ ξ ξ ξ ξ ξξ ξ
ξ ξ ξ ξ ξ ξ
⎛ ⎞′ ′⎜ ⎟⎜ ⎟′ ′⎜ ⎟=⎜ ⎟⎜ ⎟⎜ ⎟′ ′⎝ ⎠
E E E E E
E E E E EE
E E E E E
,
( )
( ) ( )
( ) ( )
2 1
2 1 2 1
2 1 2 1
|
21 1 2 | 1 |1 1 1 1 1
22 1 | 2 2 |1 1 1 1 1
var
var var
var var
s s
s
s s
s
w
M MN N N
s s sj M sj s s s s Ns ss j j s s
M MN N N
s s s s s sj M sj s Ns ss s j j s
U w w U U U U
U U U w w U U
ξ ξ
ξ ξ ξ ξ
ξ ξ ξ ξ
σ
σ
= = = = =
= = = = =
=
⎡ ⎤⎛ ⎞ ⎛ ⎞⊕ ⊕ ⊕ ⊕ ⊕⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞ ⎛ ⎞⊕ ⊕ ⊕ ⊕ ⊕⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
E
P Q Q
Q P Q
( ) ( )2 1 2 1
21 | 2 |1 1 1 1 1
var vars s
s
M MN N N
Ns s s Ns s s Ns s sj M sjs s s j jU U U U U w wξ ξ ξ ξ σ
= = = = =
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎛ ⎞ ⎛ ⎞⊕ ⊕ ⊕ ⊕ ⊕⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦⎝ ⎠
Q Q P
. Taking the expected value over 1ξ ,
( )1 2 1|
2
1 1 1
2
1 1 1
2
1 1 1
var
s s
s
s s
s
s s
s
w
M MNs
sj M sjs j j
M MNs
sj M sjs j j
M MNs
sj M sjs j j
E
w wN
w wN
w wN
ξ ξ ξ
σ
σ
σ
= = = × ×
× = = = ×
× × = = =
⎡ ⎤ =⎣ ⎦⎛ ⎡ ⎤⎛ ⎞ ⎛ ⎞⊕ ⊕ ⊕⎜ ⎢ ⎥⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎣ ⎦⎜⎜ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎜ ⊕ ⊕ ⊕⎢ ⎥⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞ ⎛ ⎞⊕ ⊕ ⊕⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦⎝
E
P 0 0
0 P 0
0 0 P
⎞⎟⎟⎟⎟
⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
⎠
or
( )1 2 1
2
| 1 1 1var
s s
s
M MNs
w N sj M sjs j jE w w
Nξ ξ ξσ
= = =
⎛ ⎞⎡ ⎤⎛ ⎞ ⎛ ⎞⎡ ⎤ = ⊗ ⊕ ⊕ ⊕⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎣ ⎦⎝ ⎠E I P .
We summarize the weighted expanded mixed model. It is given by
C06ed16.doc 5/1/2006 3:32 PM 13
( ) ( )( )11 1
N NN
w N s s N s s ws svec E vec
N ξµ µ= =
⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤= ⊗ ⊕ + ⊗ ⊕ − +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦
1Y 1 w I w U U E ,
where ( )1 2
1 1
2 21w N
N N
ENξ ξ
µµ
µ
⎛ ⎞⎜ ⎟
⎡ ⎤ ⎜ ⎟= ⊗⎢ ⎥ ⎜ ⎟⎣ ⎦⎜ ⎟⎜ ⎟⎝ ⎠
ww
Y 1 I
w
, the variance of the random effects is
( ) ( )( )1 2 11 1 1
1var1
N N N
N s s N s s N s ss s svec E vec
Nξ ξ ξµ µ µ= = =
⎡ ⎤⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎡ ⎤ ′⊗ ⊕ − = ⊗ ⊕ ⊕⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ −⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎣ ⎦I w U U P w P w ,
and ( )1 2
2
1 1 1var
s s
s
M MNs
w N sj M sjs j jw w
Nξ ξσ
= = =
⎛ ⎞⎡ ⎤⎛ ⎞ ⎛ ⎞= ⊗ ⊕ ⊕ ⊕⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦⎝ ⎠E I P .
We define the linear combination of means or totals that we want to predict as i wP ′= g Y
where ( )1 21 2 NM M N Mc c c′ ′ ′ ′ ′= ⊗g b 1 1 1 where 1sc = for all 1,...,s N= for totals, and
1s
s
cM
= for all 1,...,s N= for means and where ( )( )ib=b is an 1N × vector of constants. Of
principal interest is the linear combination that represents the total (or the mean) of PSU i ,
defined by setting i=b e where ie is an 1N × vector with all elements equal to zero, except for
element i which has the value of one. We represent either of these random variables via the
notation i i wP ′= g Y where ( )1 21 2 Ni i M M N Mc c c′ ′ ′ ′ ′= ⊗g e 1 1 1 .
We assume that the elements in the sample portion of wY will be observed, and express
iP as the sum of two parts, one which is a function of the sample, and the other which is a
function of the remaining random variables. Then, requiring the predictor to be a linear function
of the sample random variables and to be unbiased, coefficients are evaluated that minimize
C06ed16.doc 5/1/2006 3:32 PM 14
( )1 2ˆvar i iP Pξ ξ − , the expected value of the MSE. We divide the random permutation vector into a
sample and remainder, and subsequently develop the unbiased constraint.
Dividing the RP into a Sample and Remainder
We partition the random variables into the sample and remainder such that
IIw
IIII
⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
CYY
CY where
( )1 1
n N
I sIi s nNm N n NM= = × −
⎛ ⎞= ⊕⊕⎜ ⎟⎝ ⎠
C C 0 , ( ) 1 1
1 1
N N
sIN n Nm nNM i n s
II N N
sIIi s
− × = + =
= =
⎛ ⎞⊕ ⊕⎜ ⎟= ⎜ ⎟⎜ ⎟⊕⊕⎜ ⎟⎝ ⎠
0 CC
C,
( )ss s s
sI m m M m× −
⎛ ⎞= ⎜ ⎟⎝ ⎠
C I 0 and ( )s ss s s
sII M mM m m −− ×=C 0 I . Notice that with this definition,
( ) II II N
II
⎛ ⎞′ ′ =⎜ ⎟⎝ ⎠
CC C I
C.
Since ( ) ( )( )11 1
N NN
w N s s N s s ws svec E vec
N ξµ µ= =
⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤= ⊗ ⊕ + ⊗ ⊕ − +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦
1Y 1 w I w U U E ,
( ) ( )( )11 1
N NI I II N
N s s N s s ws sII II IIII
vec E vecN ξµ µ
= =
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤= ⊗ ⊕ + ⊗ ⊕ − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠
C C CY 11 w I w U U EC C CY
.
Now 1
1
1
N
I N s ssNI N N
N s ss NII
II N s ss
N N
µµ
µ
=
=
=
⎛ ⎞⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎛ ⎞ ⎝ ⎠⎡ ⎤ ⎣ ⎦⎛ ⎞ ⎜ ⎟⊗ ⊕ =⎜ ⎟⎜ ⎟ ⎢ ⎥ ⎜ ⎟⎝ ⎠ ⎡ ⎤⎣ ⎦ ⎛ ⎞⎝ ⎠ ⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠
C 1 wC 1 11 wC
C 1 w, where
( )
( )
1 1 1 1
11
11
11
N n N N
I N s s sI N s ss i s nNm N n NM s
NN
sIs Nm Nm s ssNmNN
s ssI sNm s NmnNm N n NM
NN
sI sNm Nm s
µ µ
µ
µ
= = = × − =
= × ×=×
=× = ×× −
=× × =
⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⊗ ⊕ = ⊕⊕ ⊗ ⊕⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦
⎛ ⎞⊕⎜ ⎟ ⊕⎜ ⎟⎜ ⎟ ⊕⊕⎜ ⎟=⎜ ⎟⎜ ⎟⎜ ⎟
⊕⊕⎜ ⎟⎝ ⎠
C 1 w C 0 1 w
C 0 0 w
w0 C 0 0
0 0 C
1 1
1 1
1 1
N N
sI s ss sNm
N N
sI s ss sNm
N Ns s
sI s ss sNm
µ
µ
µµ
= =×
= =×
= =×
⎛ ⎞⎛ ⎞⎜ ⎟⊕ ⊕⎜ ⎟⎜ ⎟⎛ ⎞ ⎜ ⎟⎝ ⎠⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎛ ⎞⎜ ⎟⎜ ⎟ ⊕ ⊕⎜ ⎟⎜ ⎟⎜ ⎟=⎜ ⎟ ⎝ ⎠⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎛ ⎞⎝ ⎠ ⊕ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
C w
C w
wC w
.
C06ed16.doc 5/1/2006 3:32 PM 15
and 1 1 1
N N N
sI s s sI s ss s sNm
µ µ= = =
×
⎛ ⎞⊕ ⊕ = ⊕⎜ ⎟⎜ ⎟⎝ ⎠
C w C w . Since ( )( ) ( )1 2 ss sj s s sMw w w w ′= =w , sI s sI=C w w
where ( )( ) ( )1 2 ssI sj s s smw w w w ′= =w and ( )( ) ( ), 1 , 2s s ssII sj s m s m sMw w w w+ +′= =w .
Thus, 1 1 1
N N N
sI s s sI ss s sNm
µ µ= = =
×
⎛ ⎞⊕ ⊕ = ⊕⎜ ⎟⎜ ⎟⎝ ⎠
C w w and
1 1
N N
I N s s n sI ss sµ µ
= =
⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⊗ ⊕ = ⊗ ⊕⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦C 1 w 1 w . Similarly,
( )
( )
1 1
1 1
1 1
1
1 11
1 1
1
N N
sIN NN n Nm nNM i n s
II N s s N s sN Ns s
sIIi s
N
s ssN N
NsIN n Nm nNM i n s s ss
N N
sIIi sN
s ss
µ µ
µ
µ
µ
− × = + =
= =
= =
=
− × = + ==
= =
=
⎛ ⎞⊕ ⊕⎜ ⎟⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⊗ ⊕ = ⊗ ⊕⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎜ ⎟⊕⊕⎜ ⎟⎝ ⎠
⎛ ⎞⊕⎜ ⎟⎝ ⎠
⎛ ⎞⊕ ⊕ ⎛ ⎞⎜ ⎟ ⊕⎜ ⎟= ⎜ ⎟ ⎝ ⎠⎜ ⎟⊕⊕⎜ ⎟⎝ ⎠
⎛ ⎞⊕⎜⎝ ⎠
0 CC 1 w 1 w
C
w
0 C w
C
w
1 1
1 1
N N
N n sI s ss s
N N
N sII s ss s
µ
µ
− = =
= =
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎡ ⎤⎛ ⎞⎛ ⎞⊗ ⊕ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦⎜ ⎟= ⎜ ⎟⎡ ⎤⎛ ⎞⎛ ⎞⊗ ⊕ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎝ ⎠⎣ ⎦⎝ ⎠
1 C w
1 C w.
Now
C06ed16.doc 5/1/2006 3:32 PM 16
1 1
1
1 1
1
1
N N
N n sI s ss sN
II N s s N Ns
N sII s ss s
N
N n sI s ss
N
N sII s ss
N n
µ
µµ
µ
µ
− = =
=
= =
− =
=
−
⎛ ⎞⎡ ⎤⎛ ⎞⎛ ⎞⊗ ⊕ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞ ⎜ ⎟⊗ ⊕ =⎜ ⎟⎢ ⎥ ⎜ ⎟⎡ ⎤⎝ ⎠ ⎛ ⎞⎛ ⎞⎣ ⎦ ⊗ ⊕ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎝ ⎠⎣ ⎦⎝ ⎠⎛ ⎞⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟= ⎜ ⎟⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠
⊗ ⊕
=
1 C wC 1 w
1 C w
1 C w
1 C w
11
1
N
sI ss
N
N sII ss
µ
µ
=
=
⎛ ⎞⎡ ⎤⎛ ⎞⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟
⎜ ⎟⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠
w
1 w
As a result,
1 1
1 1
1 1
N N
n sI s n sI ss s
N NI N
N n sI s N n sI ss sII
N N
N sII s N sII ss s
N
µ µ
µ µ
µ µ
= =
− −= =
= =
⎛ ⎞⎛ ⎞ ⎛ ⎞⊗ ⊕ ⊗ ⊕⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞ ⎛⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⊗ ⊕ ⊗ ⊕⎜ ⎟= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜⎜ ⎟ ⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎜ ⎟⎜ ⎟ ⎜⎝ ⎠⎜ ⎟⎜ ⎟⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⊗ ⊕ ⊗ ⊕⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝⎝ ⎠
1 w 1 w
Y 11 w 1 wY
1 w 1 w
( ) ( )( )1
I
II
vec E vecξ
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎞ ⎛ ⎞
⎡ ⎤⎜ ⎟ − +⎟ ⎜ ⎟⎣ ⎦⎜ ⎟⎟ ⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎠⎝ ⎠
EU U
E.
Also, since ( )1 2
1 1
2 21w N
N N
ENξ ξ
µµ
µ
⎛ ⎞⎜ ⎟
⎡ ⎤ ⎜ ⎟= ⊗⎢ ⎥ ⎜ ⎟⎣ ⎦⎜ ⎟⎜ ⎟⎝ ⎠
ww
Y 1 I
w
, 1 2
1
1
1
N
n sI ss
NI N
N n sI ssII
N
N sII ss
ENξ ξ
µ
µ
µ
=
− =
=
⎛ ⎞⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞⎡ ⎤⎛ ⎞ ⎛ ⎞⊗ ⊕⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟ ⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠⎝ ⎠
1 w
Y 11 wY
1 w
.
Linear Combination to Predict PSU Total or Mean We define the linear combination that we want to predict as i i wP ′= g Y where
( )1 21 2 Ni i M M N Mc c c′ ′ ′ ′ ′= ⊗g e 1 1 1 where 1sc = for all 1,...,s N= for totals, and 1s
s
cM
= for
all 1,...,s N= for means. Since ( ) II II N
II
⎛ ⎞′ ′ =⎜ ⎟⎝ ⎠
CC C I
C and II
wIIII
⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
CYY
CY,
C06ed16.doc 5/1/2006 3:32 PM 17
( ) ( )I Ii i I II w i I II
II II
P⎛ ⎞⎛ ⎞′ ′ ′ ′ ′ ′= = ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠
C Yg C C Y g C C
C Y. We express ( ) ( )i I II iI iII′ ′ ′ ′ ′=g C C g g so that
i iI I iII IIP ′ ′= +g Y g Y where ( )** 1 1 s
n N
iI n ii N s mi se c
= =
⎛ ⎞⎡ ⎤⎛ ⎞′ ′ ′ ′= ⊕ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠g 1 1 1 and
( )
1 1
1 1
nNM N n Nm N N
N NiII i II i sIIi ssIi n s
× −
= =
= + =
⎛ ⎞⎜ ⎟′ ′ ′ ′ ′= = ⊕⊕⎜ ⎟
′⊕ ⊕⎜ ⎟⎝ ⎠
0g g C g C
C. Using
( ) ( )1 21 2 1N s
N
i i M M N M i N s Msc c c c
=
⎛ ⎞′ ′ ′ ′ ′ ′ ′ ′= ⊗ = ⊗ ⊕⎜ ⎟⎝ ⎠
g e 1 1 1 e 1 1 , and defining ( )i iI iII′ ′ ′=e e e ,
( ) ( )1 1s s
N N
i iI N s M iII N s Ms sc c
= =
⎛ ⎞⎛ ⎞ ⎛ ⎞′ ′ ′ ′ ′ ′ ′= ⊗ ⊕ ⊗ ⊕⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠g e 1 1 e 1 1 , and
( )
( ) ( )
1 1
1 1
1 1 1 1 1 1s s
iII i II
nNM N n Nm N N
N Ni sIIi ssIi n s
N N N N N N
iII N s M sI i N s M sIIs i n s s i sc c
× −
= =
= + =
= = + = = = =
′ ′ ′=
⎛ ⎞⎜ ⎟′ ′= ⊕⊕⎜ ⎟
′⊕ ⊕⎜ ⎟⎝ ⎠
⎛ ⎞⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞′ ′ ′ ′ ′ ′ ′ ′= ⊗ ⊕ ⊕ ⊕ ⊗ ⊕ ⊕⊕⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎝ ⎠
g g C0
g CC
e 1 1 C e 1 1 C
.
Now
( ) ( )
( )
1 1 1 1 1
1
s s
s
N N N N N
iII N s M sI iII N s M sIs i n s s s
N
iII N s M sIs
c c
c
= = + = = =
=
⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎡ ⎤′ ′ ′ ′ ′ ′ ′ ′⊗ ⊕ ⊕ ⊕ = ⊗ ⊕ ⊕⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦ ⎝ ⎠⎛ ⎞⎡ ⎤′ ′ ′ ′= ⊗ ⊕⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠
e 1 1 C e 1 1 C
e 1 1 C
and since ( )s
s s ssI m m M m× −
⎛ ⎞= ⎜ ⎟⎝ ⎠
C I 0 , s sM sI m′ ′ ′=1 C 1 ,
( ) ( )1 1 1 1s s
N N N N
iII N s M sI iII N s ms i n s sc c
= = + = =
⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎡ ⎤′ ′ ′ ′ ′ ′ ′⊗ ⊕ ⊕ ⊕ = ⊗ ⊕⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦ ⎝ ⎠e 1 1 C e 1 1 . Also, since ( )s s
s s ssII M mM m m −− ×=C 0 I ,
( ) ( )1 1 1 1s s s
N N N N
i N s M sII i N s M ms i s sc c −= = = =
⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎡ ⎤′ ′ ′ ′ ′ ′ ′⊗ ⊕ ⊕⊕ = ⊗ ⊕⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦ ⎝ ⎠e 1 1 C e 1 1 . As a result,
C06ed16.doc 5/1/2006 3:32 PM 18
( ) ( )
( ) ( )
1 1 1 1 1 1
1 1
s s
s s s
N N N N N N
iII iII N s M sI i N s M sIIs i n s s i s
N N
iII N s m i N s M ms s
c c
c c
= = + = = = =
−= =
⎛ ⎞⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞′ ′ ′ ′ ′ ′ ′ ′ ′= ⊗ ⊕ ⊕ ⊕ ⊗ ⊕ ⊕⊕⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎝ ⎠⎛ ⎞⎛ ⎞ ⎛ ⎞⎡ ⎤ ⎡ ⎤′ ′ ′ ′ ′ ′= ⊗ ⊕ ⊗ ⊕⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎝ ⎠
g e 1 1 C e 1 1 C
e 1 1 e 1 1
or ( ) ( )1 1s s s
N N
iII iII N s m i N s M ms sc c −= =
⎛ ⎞⎛ ⎞ ⎛ ⎞⎡ ⎤ ⎡ ⎤′ ′ ′ ′ ′ ′ ′= ⊗ ⊕ ⊗ ⊕⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎝ ⎠g e 1 1 e 1 1 .
With these definitions, ( ) Ii i w iI iII
II
P⎛ ⎞
′ ′ ′= = ⎜ ⎟⎝ ⎠
Yg Y g g
Y, where
( )** 1 1 s
n N
iI n ii N s mi se c
= =
⎛ ⎞⎡ ⎤⎛ ⎞′ ′ ′ ′= ⊕ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠g 1 1 1 and
( ) ( )* ** 1 1 * 1 1s s s
N N N N
iII N n ii N s m N ii N s M mi n s i se c e c− −= + = = =
⎛ ⎞⎛ ⎞ ⎛ ⎞⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞′ ′ ′ ′ ′ ′ ′= ⊕ ⊕ ⊕ ⊕⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎝ ⎠g 1 1 1 1 1 1 .
Unbiased Constraint for Expanded Model
We require the predictor of i iI I iII IIP ′ ′= +g Y g Y to be a linear function of the sample
random variables, i IP ′= L Y , and to be unbiased. The unbiased constraint requires that
( ) ( )1 2 1 2
ˆ 0Ii i iI iII
II
E P P Eξ ξ ξ ξ
⎛ ⎞′ ′ ′− = − − =⎜ ⎟
⎝ ⎠
YL g g
Y. Since
1 2
1
1
1
N
n sI ss
NI N
N n sI ssII
N
N sII ss
ENξ ξ
µ
µ
µ
=
− =
=
⎛ ⎞⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞⎡ ⎤⎛ ⎞ ⎛ ⎞⊗ ⊕⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟ ⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠⎝ ⎠
1 w
Y 11 wY
1 w
,
this implies that ( )( )
1
1
1
0
N
n sI ss
NN
N n sI siI iII s
N
N sII ss
N
µ
µ
µ
=
− =
=
⎡ ⎤⎛ ⎞⎛ ⎞⊗ ⊕⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎛ ⎞⎡ ⎤⎛ ⎞⊗ ⊕′ ′ ′⎢ ⎥⎜ ⎟− − =⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎡ ⎤⎛ ⎞⊗ ⊕⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎣ ⎦⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦
1 w
11 wL g g
1 w
or equivalently that
C06ed16.doc 5/1/2006 3:32 PM 19
( )1
1
1
0
N
N n sI ssN
iI n sI s iII Ns
N sII ss
µ
µµ
− =
=
=
⎛ ⎞⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞⎛ ⎞ ⎜ ⎟′ ′ ′− ⊗ ⊕ − =⎜ ⎟⎜ ⎟ ⎜ ⎟⎡ ⎤⎝ ⎠ ⎛ ⎞⎝ ⎠ ⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠
1 wL g 1 w g
1 w.
For this expression to equal zero,
1
1 1
1
0
N
N n sI ssN N
n sI s iI n sI s iII Ns s
N sII ss
µ
µ µµ
− =
= =
=
⎛ ⎞⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜ ⎟′ ′ ′⊗ ⊕ − ⊗ ⊕ − =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎡ ⎤⎝ ⎠ ⎝ ⎠ ⎛ ⎞⎝ ⎠ ⎝ ⎠ ⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠
1 wL 1 w g 1 w g
1 w
for all values of sµ , 1,...,s N= .
We use the definition of ( )( ) ( )1 2i n′′ ′ ′= =L L L L L where
( )( ) ( )1 2 sis isj is is ismL L L L ′= =L , and ( )( ) ( )1 2i is i i iN′′ ′ ′= =L L L L L to simplify this
expression. First, notice that
( )
11 1
21 11 21
1 1
N N
sI s sI ss sN N
NsI s sI ss sn sI s ns
N N
sI s n sI ss s
µ µ
µ µµ
µ µ
= =
= ==
= =
⎛ ⎞ ⎛ ⎞′⊕ ⊕⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟′⊕ ⊕⎛ ⎞⎛ ⎞′ ′ ′ ′⊗ ⊕ = =⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟′⊕ ⊕⎝ ⎠ ⎝ ⎠
w L w
w L wL 1 w L L L
w L w
and
( )( )
1 21 1
1 1 1 2 2 2
N N
i sI s i i iN sI ss s
i I i I iN NI N
µ µ
µ µ µ= =
′ ′ ′ ′⊕ = ⊕
′ ′ ′=
L w L L L w
L w L w L w
C06ed16.doc 5/1/2006 3:32 PM 20
where ( )1
21 2
1
s
s
s
s
ms
is sI s is is ism s s isj sjj
sm
ww
L L L L w
w
µ µ µ=
⎛ ⎞⎜ ⎟ ⎛ ⎞⎜ ⎟′ = = ⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎜ ⎟⎝ ⎠
∑L w . Let us define
1
sm
s wisI isj sjj
m L L w=
= ∑ . Then is sI s s wisI sm Lµ µ′ =L w . As a result,
1 1 1 2 2 21 1 1 1
n n nN
n sI s wi I wi I N wiNI Ns i i i
m L m L m Lµ µ µ µ=
= = =
⎛ ⎞⎛ ⎞⎛ ⎞′ ⊗ ⊕ =⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠∑ ∑ ∑L 1 w .
We simplify the expressions for 1
N
iI n sI ssµ
=
⎛ ⎞⎛ ⎞′ ⊗ ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠g 1 w and
1
1
N
N n sI ss
iII N
N sII ss
µ
µ
− =
=
⎛ ⎞⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟′ ⎜ ⎟⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠
1 wg
1 w
next. Since ( )** 1 1 s
n N
iI n ii N s mi se c
= =
⎛ ⎞⎡ ⎤⎛ ⎞′ ′ ′ ′= ⊕ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠g 1 1 1 ,
( )
( )
*1 * 1 1 1
1
1** 1 1
1
s
s
N n N N
iI n sI s n ii N s m n sI ss i s s
N
sI ss
Nn N
sI ssn ii N s mi s
N
sI ss
n
e c
e c
e
µ µ
µ
µ
µ
= = = =
=
== =
=
⎛ ⎞⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞′ ′ ′ ′⊗ ⊕ = ⊕ ⊕ ⊗ ⊕⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎝ ⎠⎛ ⎞⊕⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞ ⊕⎡ ⎤⎛ ⎞′ ′ ′= ⊕ ⊕ ⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⊕⎝ ⎠
′=
g 1 w 1 1 1 1 w
w
w1 1 1
w
1
( )
( )
( )
1 1 1
2 1 1
1 1
s
s
s
N N
i N s m sI ss s
N N
i N s m sI ss s
N N
in N s m sI ss s
c
e c
e c
µ
µ
µ
= =
= =
= =
⎛ ⎞⎛ ⎞′ ′⊕ ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞′ ′⊕ ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟
⎛ ⎞⎜ ⎟′ ′⊕ ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
1 1 w
1 1 w
1 1 w
or
C06ed16.doc 5/1/2006 3:32 PM 21
( )
( )
( )
( )1 2
1 1 1
2 1 11
1 1
1 1 1 1 2 2 2
2
s
s
s
N
N N
i N s m sI ss s
N NN
i N s m sI ss siI n sI s ns
N N
in N s m sI ss s
i m I m I N m NI N
in
e c
e c
e c
e c c c
e c
µ
µµ
µ
µ µ µ
= =
= ==
= =
⎛ ⎞⎛ ⎞′ ′⊕ ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞′ ′⊕ ⊕⎛ ⎞ ⎜ ⎟⎛ ⎞ ⎜ ⎟′ ′⊗ ⊕ = ⎝ ⎠⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎜ ⎟⎜ ⎟
⎛ ⎞⎜ ⎟′ ′⊕ ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
′ ′ ′
′=
1 1 w
1 1 wg 1 w 1
1 1 w
1 w 1 w 1 w
1 ( )
( )
1 2
1 2
1 1 1 2 2 2
1 1 1 2 2 2
N
N
m I m I N m NI N
in m I m I N m NI N
c c
e c c c
µ µ µ
µ µ µ
⎛ ⎞⎜ ⎟⎜ ⎟′ ′ ′⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟′ ′ ′⎝ ⎠
1 w 1 w 1 w
1 w 1 w 1 w
or ( )1 2* 1 1 1 2 2 21 * 1N
nN
iI n sI s ii m I m I N m NI Ns i
e c c cµ µ µ µ=
=
⎛ ⎞⎛ ⎞′ ′ ′ ′⊗ ⊕ =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∑g 1 w 1 w 1 w 1 w .
Also, 1
1
N
N n sI ss
iII N
N sII ss
µ
µ
− =
=
⎛ ⎞⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟′ ⎜ ⎟⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠
1 wg
1 wwhere
( ) ( )* ** 1 1 * 1 1s s s
N N N N
iII N n ii N s m N ii N s M mi n s i se c e c− −= + = = =
⎛ ⎞⎛ ⎞ ⎛ ⎞⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞′ ′ ′ ′ ′ ′ ′= ⊕ ⊕ ⊕ ⊕⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎝ ⎠g 1 1 1 1 1 1 . Then
( )
( )
1
** 1 1 1
1
** 1 1
s
s s
N
N n sI ss N N N
iII N n ii N s m N n sI sN i n s s
N sII ss
N N
N ii N s M mi s
e c
e c
µ
µµ
− =
− −= + = =
=
−= =
⎛ ⎞⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ ⎛ ⎞ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎜ ⎟′ ′ ′ ′= ⊕ ⊕ ⊗ ⊕⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎜ ⎟⎡ ⎤ ⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎣ ⎦⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠⎛ ⎡ ⎤⎛ ⎞′ ′ ′+ ⊕ ⊕⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎝
1 wg 1 1 1 1 w
1 w
1 1 1
( ) ( )
( )
1
* *1 1 1 1* 1 * 1
* *1 1* 1
s s s
s s s
N
N sII ss
N NN N N N
ii N s m sI s ii N s M m sII ss s s si n i
N N N
ii N s m sI s ii N s M m sIIs si n
e c e c
e c e c
µ
µ µ
µ
=
−= = = == + =
−= == +
⎞ ⎡ ⎤⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟ ⎢ ⎥⎢ ⎥⎝ ⎠⎣ ⎦⎠ ⎣ ⎦⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞′ ′ ′ ′= ⊕ ⊕ + ⊕ ⊕⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠
⎛ ⎞′ ′ ′ ′= ⊕ + ⊕⎜ ⎟⎝ ⎠
∑ ∑
∑
1 w
1 1 w 1 1 w
1 1 w 1 1 w( )* 1
N
si
µ=
⎛ ⎞⎜ ⎟⎝ ⎠
∑or
C06ed16.doc 5/1/2006 3:32 PM 22
( ) ( )
( )1 2
1
* *1 1* 1 * 1
1
* 1 1 1 2 2 2* 1
s s s
N
N
N n sI ss N NN N
iII ii N s m sI s ii N s M m sII sN s si n iN sII ss
N
ii m I m I N m NI Ni n
i
e c e c
e c c c
e
µ
µ µµ
µ µ µ
− =
−= == + =
=
= +
⎛ ⎞⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ ⎛ ⎞ ⎛ ⎞⎜ ⎟′ ′ ′ ′ ′= ⊕ + ⊕⎜ ⎟ ⎜ ⎟⎜ ⎟⎡ ⎤ ⎝ ⎠ ⎝ ⎠⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠
′ ′ ′= +
∑ ∑
∑
1 wg 1 1 w 1 1 w
1 w
1 w 1 w 1 w
( )1 1 2 2* 1 1 1 2 2 2* 1
N N
N
i M m II M m II N M m NII Ni
c c cµ µ µ− − −=
′ ′ ′∑ 1 w 1 w 1 w
.
We use these expressions to simplify the unbiased constraint given by
1
1 1
1
0
N
N n sI ssN N
n sI s iI n sI s iII Ns s
N sII ss
µ
µ µµ
− =
= =
=
⎛ ⎞⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜ ⎟′ ′ ′⊗ ⊕ − ⊗ ⊕ − =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎡ ⎤⎝ ⎠ ⎝ ⎠ ⎛ ⎞⎝ ⎠ ⎝ ⎠ ⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠
1 wL 1 w g 1 w g
1 w.
. Since 1 1 1 2 2 21 1 1 1
n n nN
n sI s wi I wi I N wiNI Ns i i i
m L m L m Lµ µ µ µ=
= = =
⎛ ⎞⎛ ⎞⎛ ⎞′ ⊗ ⊕ =⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠∑ ∑ ∑L 1 w ,
( )1 2* 1 1 1 2 2 21 * 1N
nN
iI n sI s ii m I m I N m NI Ns i
e c c cµ µ µ µ=
=
⎛ ⎞⎛ ⎞′ ′ ′ ′⊗ ⊕ =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∑g 1 w 1 w 1 w 1 w , and
( )
( )
1 2
1 1 2 2
1
* 1 1 1 2 2 2* 1
1
* 1 1 1 2 2 2* 1
N
N N
N
N n sI ss N
iII ii m I m I N m NI NNi n
N sII ss
N
ii M m II M m II N M m NII Ni
e c c c
e c c c
µ
µ µ µµ
µ µ µ
− =
= +
=
− − −=
⎛ ⎞⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟′ ′ ′ ′= +⎜ ⎟⎡ ⎤⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠
′ ′ ′
∑
∑
1 wg 1 w 1 w 1 w
1 w
1 w 1 w 1 w
,
the unbiased constraint is given by
C06ed16.doc 5/1/2006 3:32 PM 23
( )
( )
( )
1 2
1 2
1 1 2 2
1 1 1 2 2 21 1 1
* 1 1 1 2 2 2* 1
* 1 1 1 2 2 2* 1
* 1 1 1 2 2 2* 1
N
N
N N
n n n
wi I wi I N wiNI Ni i i
n
ii m I m I N m NI Ni
N
ii m I m I N m NI Ni n
N
ii M m II M m II N M m NII Ni
m L m L m L
e c c c
e c c c
e c c c
µ µ µ
µ µ µ
µ µ µ
µ µ µ
= = =
=
= +
− − −=
⎛ ⎞ +⎜ ⎟⎝ ⎠
′ ′ ′− +
′ ′ ′− +
′ ′ ′−
∑ ∑ ∑
∑
∑
1 w 1 w 1 w
1 w 1 w 1 w
1 w 1 w 1 w 0=∑
or
( )
( )
( )
1 2
1 2
1 1 2 2
1 1 1 2 2 21 1 1
* 1 1 1 2 2 2* 1
* 1 1 1 2 2 2* 1
* 1 1 1 2 2 2* 1
N
N
N N
n n n
wi I wi I N wiNI Ni i i
n
ii m I m I N m NI Ni
N
ii m I m I N m NI Ni n
n
ii M m II M m II N M m NII Ni
m L m L m L
e c c c
e c c c
e c c c
µ µ µ
µ µ µ
µ µ µ
µ µ µ
= = =
=
= +
− − −=
⎛ ⎞ +⎜ ⎟⎝ ⎠
′ ′ ′− +
′ ′ ′− +
′ ′ ′−
∑ ∑ ∑
∑
∑
1 w 1 w 1 w
1 w 1 w 1 w
1 w 1 w 1 w
( )1 1 2 2* 1 1 1 2 2 2* 1
0N N
N
ii M m II M m II N M m NII Ni n
e c c cµ µ µ− − −= +
+
′ ′ ′− =
∑
∑ 1 w 1 w 1 w
or
1 2
1 1 2 2
*1 *1 2 *2
1 21 2* 1 * 1 * 1* 1 * 2 *
1 2
*
N
N N
N wi NIwi I wi In n n
m NIm I m I Ni i iii ii ii N
M m II M m II M m NII
ii
m Lm L m L
e c e c e c
e c
µ µ µ= = =
− − −
⎛ ⎞⎡ ⎤⎡ ⎤ ⎡ ⎤⎜ ⎟⎢ ⎥⎢ ⎥ ⎢ ⎥ ′′ ′ ⎛ ⎞⎛ ⎞ ⎛ ⎞ +⎜ ⎟⎢ ⎥⎢ ⎥ ⎢ ⎥
− − − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ′ ′ ′ ⎟+ + +⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎝ ⎠
−
∑ ∑ ∑ 1 w1 w 1 w
1 w 1 w 1 w
1 2
1 1 2 2
1 21 1 * 2 2 *
* 1 * 1 * 11 2
0N
N N
N n n m NIm I m Iii ii N N
i n i iM m II M m II M m NII
e c e cµ µ µ= + = =− − −
⎛ ⎞′′ ′ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟− − =⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ′ ′ ′ ⎟+ + +⎢ ⎥⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎝ ⎠∑ ∑ ∑
1 w1 w 1 w
1 w 1 w 1 w
. In order for this expression to be equal to zero for all values of sµ , 1,...,s N= , we require
* * ** 1 * 1
0s s
s s s s
n Nm sI m sIs wi sI ii s ii s
i i nM m sII M m sII
m L e c e c= = +− −
′ ′⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞− − =⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟′ ′+ +⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦
∑ ∑1 w 1 w
1 w 1 w for all 1,...,s N= . Now
s wisI is sIm L ′= L w . Then the unbiased constraint is given by
C06ed16.doc 5/1/2006 3:32 PM 24
( ) ( )* * ** 1 * 1
0s s s s s s
n N
i s sI ii s m sI M m sII ii s m sI M m sIIi i n
e c e c− −= = +
⎡ ⎤ ⎡ ⎤′ ′ ′ ′ ′− + − + =⎣ ⎦ ⎣ ⎦∑ ∑L w 1 w 1 w 1 w 1 w .
When i n≤ , * 1iie = when *i i= , and zero otherwise. As a result, the unbiased constraint when
i n≤ is given by ( )** 1
s s s
n
i s sI s m sI M m sIIi
c −=
⎛ ⎞′ ′ ′= +⎜ ⎟⎝ ⎠∑L w 1 w 1 w . When i n> , * 1iie = when *i i= , and zero
otherwise. As a result, the unbiased constraint when i n> is given by
[ ] ( )** 1
s s s
n
i s sI s m sI M m sIIi
c −=
′ ′ ′= +∑ L w 1 w 1 w .
We consider this constraint in more detail. First, notice that s wisI is sIm L ′= L w where
1
1 sm
wisI isj sjjs
L L wm =
= ∑ . Then
[ ]* ** 1 * 1
** 1 1
*1 * 1
s
s
n n
i s sI s wi sIi i
mn
i sj sji j
m n
sj i sjj i
m L
L w
w L
= =
= =
= =
′ =
=
⎛ ⎞= ⎜ ⎟
⎝ ⎠
∑ ∑
∑∑
∑ ∑
L w
.
Also, ( )1
s
s s s
M
s m sI M m sII s sjj
c c w−=
′ ′+ = ∑1 w 1 w . Let ** 1
n
sj i sji
nL L=
= ∑ . Then the unbiased constraint is given
by 1 1
s sm M
sj sj s sjj j
n w L c w= =
=∑ ∑ . We express this as
1
2s
s
s
s ssI M s
sm
LL c
nL
⎛ ⎞⎜ ⎟⎜ ⎟′ ′=⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
w 1 w .
TO HERE 5/1/2006
C06ed16.doc 5/1/2006 3:32 PM 25
Let us define 1
1 sm
is isjjs
L Lm =
= ∑ . Now 1
1 1s
N
I nNm n msN N =
⎛ ⎞⎛ ⎞= ⊗ ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠X 1 1 1 . Then
1 1
1 1n N
nNm s isi s
m LN N= =
⎛ ⎞′ = ⎜ ⎟⎝ ⎠
∑ ∑L 1 , and
( )
( )
( )
1 2
1 21 1
11
1 21
1 1 2 21
1 1
1
1
1
s s
s
N
N N
n m n n ms s
n N
i msi
n
i m i m iN mi
n
i i N iNi
N N
N
N
m L m L m LN
= =
==
=
=
⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞′ ′ ′ ′⊗ ⊕ = ⊗ ⊕⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎡ ⎤⎛ ⎞′= ⊕⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
′ ′ ′=
=
∑
∑
∑
L 1 1 L L L 1 1
L 1
L 1 L 1 L 1.
As a result,
( )1 1 2 2
1 1 1
1 1 2 21 1 1 1 1
1 1
1
n N n
I s is i i N iNi s i
N n n n n
s is i i N iNs i i i i
m L m L m L m LN N
m L m L m L m LN
= = =
= = = = =
⎛ ⎞⎛ ⎞′ = ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎛ ⎞⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎝ ⎠
∑ ∑ ∑
∑ ∑ ∑ ∑ ∑
L X.
In addition, 1
1 1s
N
N N MsN N =
⎛ ⎞⎛ ⎞⎛ ⎞= ⊗ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠X 1 1 1 and
( )1 21 2 Ni i M M N Mc c c⎡ ⎤′ ′ ′ ′ ′= ⊗⎣ ⎦g e 1 1 1 so that
( )
[ ] ( ) ( )
1 2
1 2 1 2
1 2 1
1 2 1 2 1
1 1
1 1
N s
N N s
N
i i M M N M N N Ms
N
i N M M N M i N M M N M Ms
c c cN N
c c c c c cN N
=
=
⎛ ⎞⎛ ⎞⎛ ⎞⎡ ⎤′ ′ ′ ′ ′= ⊗ ⊗ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎣ ⎦ ⎝ ⎠⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎡ ⎤⎛ ⎞⎡ ⎤′ ′ ′ ′ ′ ′ ′ ′= ⊗ ⊗ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦⎝ ⎠⎝ ⎠
g X e 1 1 1 1 1 1
e 1 1 1 1 1 e 1 1 1 1 1
. Now 1i N′ =e 1 , ( )1 21 21
N
N
M M N M s ss
c c c c M=
′ ′ ′ = ∑1 1 1 1 and
C06ed16.doc 5/1/2006 3:32 PM 26
( ) ( )
( )
1 1 1
2 2 2
1 2 1 21 2 1 21
1 1 2 2
N s N
N N N
M M M
N M M MM M N M M M M N Ms
M M M
N N
c c c c c c
c M c M c M
=
⎛ ⎞⎜ ⎟⎜ ⎟⎛ ⎞′ ′ ′ ′ ′ ′⊕ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎜ ⎟⎜ ⎟⎝ ⎠
=
1 0 0
0 1 01 1 1 1 1 1 1
0 0 1.
Then ( )1 1 2 21
1 1N
i s s N Ns
c M c M c M c MN N=
⎛ ⎞′ = ⎜ ⎟⎝ ⎠∑g X .
Using these expressions, the unbiased constraint is given by
1 1N NI i
N N N N
′ ′⎛ ⎞ ⎛ ⎞′ ′=⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
0 0L X g X
0 P 0 P or
( )
1 1 2 21 1 1 1 1
1 1 2 21
11
11
N n n n nN
s is i i N iNs i i i i N N
NN
s s N Ns N N
m L m L m L m LN
c M c M c M c MN
= = = = =
=
′⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞=⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠′⎛ ⎞⎛ ⎞
⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑ ∑ ∑ ∑ ∑
∑
00 P
00 P
or
( )1 1 2 2 1 1 2 21 1 1 1 1 1
N n n n n N
s is i i N iN N s s N N Ns i i i i s
m L m L m L m L c M c M c M c M= = = = = =
⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞=⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠∑ ∑ ∑ ∑ ∑ ∑P P
. We express this as
1 1 1 1 2 2 2 21 1 1 1 1
N n n n n
s is s s i i N iN N N Ns i i i i
m L c M m L c M m L c M m L c M= = = = =
⎛ ⎞⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− − − − =⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎝ ⎠⎝ ⎠
∑ ∑ ∑ ∑ ∑ P 0
.
Now 1 11 1 1 1 1 1
1 1 1
n n
i ii i
c Mm L c M m Lm= =
⎛ ⎞− = −⎜ ⎟
⎝ ⎠∑ ∑ . The unbiased constraint implies that for all
1,...,s N= , 1
0n
s sis
i s
c MLm=
− =∑ or s ss
s
c MLnm
= .
Predictors that are Unbiased
C06ed16.doc 5/1/2006 3:32 PM 27
We consider now a predictor that is unbiased. First, let us express the vector IY . This
vector is given by ( )( ) ( )1 2I iI I I nI′′ ′ ′= =Y Y Y Y Y where
( )( ) ( )1 1 2 2iI is sI i I i I iN NIU U U U ′′ ′ ′= =Y Y Y Y Y , ( )( ) ( )1 2 ssI sj s s smY Y Y Y ′= =Y , and
( )
1
sMs
sj jt stt
Y U y=
= ∑ . Using the expression for ( )( ) ( )1 2i n′′ ′ ′= =L L L L L where
( )( ) ( )1 2 sis isj is is ismL L L L ′= =L , and ( )( ) ( )1 2i is i i iN′′ ′ ′= =L L L L L ,
( )1
21 2
1
In
II n i iI
i
nI
=
⎛ ⎞⎜ ⎟⎜ ⎟′ ′ ′ ′ ′= =⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
∑
YY
L Y L L L L Y
Y
.
Also, ( )1 1
2 21 2
1
i IN
i Ii iI i i iN is is sI
s
iN NI
UU
U
U=
⎛ ⎞⎜ ⎟⎜ ⎟′ ′ ′ ′ ′= =⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
∑
YY
L Y L L L L Y
Y
and ( )1
21 2
1
s
s
s
sm
sis sI is is ism isj sj
j
sm
YY
L L L L Y
Y=
⎛ ⎞⎜ ⎟′ ⎜ ⎟′ = =⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
∑L Y .
As a result, 1 1 1
smn N
I is isj sji s j
U L Y= = =
′ = ∑∑ ∑L Y . Recall that the unbiased constraint is given by
1 1
smn
isj s si j
L c M= =
=∑∑ . We express ( )1 1 1
smN n
I isj is sjs i j
L U Y= = =
⎡ ⎤′ = ⎢ ⎥
⎣ ⎦∑ ∑∑L Y . Notice that when expressed in the
expanded form, the unbiased constraint does not require probability proportional to size
sampling.
C06ed16.doc 5/1/2006 3:32 PM 28
Unbiased Constraint in Terms of Positions Corresponding to a Random Permutation of the
Population.
The unbiased constraint is given by 1 ss
s
cLn f⎛ ⎞
= ⎜ ⎟⎝ ⎠
which we can express as
1 1
1 1smns
isji js s
cLnm n f= =
⎛ ⎞= ⎜ ⎟
⎝ ⎠∑∑ or
1 1
smn
isj s si j
L c M= =
=∑∑ . Notice that isjL is a function of s , the cluster. We
consider here the unbiased constraint when the expanded random variables are collapsed to
random variables that only depend on position. In a population with equal size clusters, and
equal size samples per cluster, random variables are usually represented as 1
N
ij is sjs
Y U Y=
=∑ . This
notation is problematic when clusters and sample sizes per cluster differ.
We illustrate the problem with the notation, 1
N
ij is sjs
Y U Y=
=∑ when clusters differ in size.
Suppose 3N = , 1 10M = , 2 6M = and 3 4M = , and we select a sample of 50% of the units in
each cluster. The sample sizes for the clusters are 1 5m = , 2 3m = and 3 2m = . Suppose further
that we select a simple random sample of 2n = clusters. In this setting, we represent the sample
in expanded form as
C06ed16.doc 5/1/2006 3:32 PM 29
11 11
11 12
11 13
11 14
11 15
12 21
12 22
12 2311 1
13 3112 2
13 31
2 21 1
22 2
23 3
I
I
II
II I
I
I
U YU YU YU YU Y
U YU YU YUU YUUU
UUU
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞⎛ ⎞ ⎝ ⎠′
⎜ ⎟⎜ ⎟′⎜ ⎟⎜ ⎟
⎜ ⎟⎜ ⎟′⎛ ⎞ ⎝ ⎠⎜ ⎟= = =⎜ ⎟ ⎜ ⎟⎛ ⎞′⎝ ⎠ ⎜ ⎟⎜ ⎟⎜ ⎟′⎜ ⎟⎜ ⎟⎜ ⎟′⎜ ⎟⎝ ⎠⎝ ⎠
YYYY
YY Y
YY
13 31
21 11
21 12
21 13
21 14
21 15
22 21
22 22
22 23
23 31
23 31
Y
U YU YU YU YU Y
U YU YU Y
U YU Y
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠⎟⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
.
Suppose we try to express this in terms of the random variables 1
N
ij is sjs
Y U Y=
=∑ . Without a careful
examination of the random variables, it is tempting to represent the sample as
C06ed16.doc 5/1/2006 3:32 PM 30
( )( ) 1
2
11
11
1
21
21
2
m
ij
m
YY
YY
YY
Y
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
= ⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
where im in the terms iimY represents the number of units selected in PSU i .
This notation is problematic since im is a random variable, and hence the dimensions of the
vectors are random.
Other aspects are also problematic. For example, consider the representation of SSU
5j = in a sample where the realized cluster for PSU=2 is cluster 1. It is perfectly clear to
express this random variable as 21 15U Y . However, the notation 25ijY Y= is not immediately clear
without some more definitions. To see this, notice that
25 2 5 21 15 22 25 23 351
N
ij s ss
Y Y U Y U Y U Y U Y=
= = = + +∑ . In this expression, the term ( )3
335 5 3
1
M
t tt
Y U y=
= ∑
requires definition of the terms ( )35tU for 31,..., 4t M= = . In words, this random variable is an
indicator random variable that has a value of 1 if SSU 5 corresponds to unit t . Since there are
only 3 4M = units in cluster 3, the random variable ( )35tU has value 0 with probability 1. The
need for such seemingly artificial random variables detracts from the simplicity of the ijY
notation.
Unbiased Constraint in Terms of Positions Corresponding to a Single Random Variable
per Cluster.
C06ed16.doc 5/1/2006 3:32 PM 31
The unbiased constraint is given by 1 ss
s
cLn f⎛ ⎞
= ⎜ ⎟⎝ ⎠
which we can express as
1 1
1 1smns
isji js s
cLnm n f= =
⎛ ⎞= ⎜ ⎟
⎝ ⎠∑∑ or
1 1
smn
isj s si j
L c M= =
=∑∑ . Notice that isjL is a function of s , the cluster. We
consider here the unbiased constraint when the expanded random variables are collapsed to
random variables with a single random variable per cluster in the sample, and in the remainder.
We first outline the basic idea in an example for a single cluster. Subsequently, we develop the
idea for the expanded population, and examine the implication for the unbiased constraint.
Suppose that we have a vector of random variables given by I′L Y . Now, we define
( )ss s s
sI m m M m× −
⎛ ⎞= ⎜ ⎟⎝ ⎠
C I 0 and ( )1 1
n N
I sIi s nNm N n NM= = × −
⎛ ⎞= ⊕⊕⎜ ⎟⎝ ⎠
C C 0 so that I I=Y C Y . We denote
( )
1
sMs
sj jt stt
Y U y=
= ∑ . Let ( )( ) ( )1 2 ss sj s s sMY Y Y Y ′= =Y and
( )( ) ( )1 2 ssI sj s s smY Y Y Y ′= =Y , while ( )( ) ( ) ( )( )1 2 ss ssII sj sMs m s mY Y Y Y+ +
′= =Y . As a
result, ( )s sI sII′′ ′=Y Y Y . Next, let us represent ( )( ) ( )1 1 2 2i is s i i iN NU U U U ′′ ′ ′= =Y Y Y Y Y or
( )i is sI sIIU⎛ ⎞⎛ ⎞′′ ′= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
Y Y Y . Finally, ( )( ) ( )1 2i N′′ ′ ′ ′= =Y Y Y Y Y .
The expression for I I=Y C Y can be simplified. Thus, since
( )
( )
( )
111
1
22 11
1 1
NN
sIssIs Nm Nm Nm Nm Nm N n NMNN
sINm N n NMsI sNm Nm s Nm NmI
N N NNm N n NM
sI sI nNm Nm Nm Nm s s
== × × × −
× − =× = ×
× −× × = =
⎛ ⎛ ⎞⊕⎛ ⎞ ⎜ ⎟⎜⊕ ⎝ ⎠⎜ ⎟ ⎜⎛ ⎞⎜ ⎟ ⎜⎜ ⎟ ⎛ ⎞⎜ ⎟ ⊕⊕ ⎜ ⎟⎜ ⎟= = ⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟ ⎛ ⎞⊕ ⊕⎜ ⎟⎝ ⎠ ⎝ ⎠⎝
C YC 0 0 0Y
0 C YY0 C 0C Y
Y00 0 C C Y
⎞⎟⎟⎟
⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
⎠
,
C06ed16.doc 5/1/2006 3:32 PM 32
and
2
1
1 2
1 11 1 1
1 11 1 1
2 22 2 2 21 1
2 21
1 1
N
N
I Ii i I
II III M M
I IN I i i IM M
II IIsI is
NIM MNI
iNNII
U U
U U
U
× ×
× ×
=
× ×
⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞⎜ ⎟⎜ ⎟ ⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜⎛ ⎞ ⎜ ⎟⎜ ⎟⊕ = =⎝ ⎠ ⎝⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎛ ⎞⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
Y YC
Y YC 0 0Y Y0 C 0 CY YC Y
0 0 CYY
NIiN NI
NII
U
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
YC
Y
where
sIis sI is sI
sII
U U⎛ ⎞
=⎜ ⎟⎝ ⎠
YC Y
Y, then ( )( )
1 1
2 2
1
i I
Ni I
sI i is sIs
iN NI
UU
U
U=
⎛ ⎞⎜ ⎟
⎛ ⎞ ⎜ ⎟⊕ = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎜ ⎟⎜ ⎟⎝ ⎠
YY
C Y Y
Y
and hence
11 1
12 2
1
21 1
22 2
2
1 1
2 2
I
I
N NI
I
I
I I
N NI
n I
n I
nN NI
UU
U
UU
U
UU
U
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= = ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
YY
Y
YY
Y C YY
YY
Y
.
Recall that the predictor is a linear function of IY given by I′L Y where we have
determined the unbiased constraint to be such that 1 ss
s
cLn f⎛ ⎞
= ⎜ ⎟⎝ ⎠
or 1 1
1 1smns
isji js s
cLnm n f= =
⎛ ⎞= ⎜ ⎟
⎝ ⎠∑∑ for all
1,...,s N= , or equivalently, that [ ]I iI I iII II′ ′ ′− + =L X g X g X 0 . What is the corresponding linear
C06ed16.doc 5/1/2006 3:32 PM 33
constraint evaluated on a collapsed set of random variables corresponding either to the total or a
weighted value of the sample PSU units?
Summary of Linear Constraint for Expanded Model
The expanded model is given by
( ) ( )1 1
1 1s
N N
N N Mi sN Nµ
= =
⎡ ⎤⎡ ⎤ ⎛ ⎞= + ⊗ + ⊕⊕ + +⎜ ⎟⎢ ⎥⎢ ⎥ ⎝ ⎠⎣ ⎦ ⎣ ⎦Y 1 1 Z β 1 M B E
or
( )I I I I
II II II II
µ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞= + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
Y X Z EM B
βY X Z E
where, 1
1 1s
N
I nNm n msN N =
⎛ ⎞⎛ ⎞= ⊗ ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠X 1 1 1 ,
( )
( ) ( ) ( )
1
1
1 1
1 1
s
s s
N
N n mN n Nm s
II N
N M mN N M m s
N N
N N
−− =
−− =
⎛ ⎞⎛ ⎞⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎜ ⎟= ⎜ ⎟⎛ ⎞⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠
1 1 1X
1 1 1,
( )1 1 s
n N
I mi s nNm N n N= = × −
⎛ ⎞= ⊕⊕⎜ ⎟⎝ ⎠
Z 1 0 and ( ) 1 1
1 1
s
s s
N N
mN n Nm nN i n s
II N N
M mi s
− × = + =
−= =
⎛ ⎞⊕ ⊕⎜ ⎟= ⎜ ⎟⎜ ⎟⊕⊕⎜ ⎟⎝ ⎠
0 1Z
1. We require the predictor of
i iI I iII IIP ′ ′= +g Y g Y to be a linear function of the sample random variables, i IP ′= L Y , and to be
unbiased. This implies that ( ) ( )ˆ 0Ii i iI iII
II
E P P E⎛ ⎞
′ ′ ′− = − − =⎜ ⎟⎝ ⎠
YL g g
Y. Since
I I
II II
Eµ⎛ ⎞ ⎛ ⎞⎛ ⎞
=⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
Y XβY X
, this implies that I iI I iII II⎡ ⎤′ ′ ′− + =⎣ ⎦L X g X g X 0 .
Collapsing Random Variables to Sample and Remainder PSU Totals
Consider an example where we collapse the random variables in IY to a simple total for
each PSU in the sample, and a total for the PSU in the remainder. Let us define
C06ed16.doc 5/1/2006 3:32 PM 34
( )
( )
( ) ( )
( ) ( )2
n Nm n N NM nm
I N n Nm N n NN M m
IIN n nNm N N M mN Nm N n
× −
−− × −
− ×−× −
′⊗⎛ ⎞⎜ ⎟⎜ ⎟′⎛ ⎞ ′⊗⎛ ⎞′ = =⎜ ⎟ ⎜ ⎟⎜ ⎟′⎝ ⎠ ⎜ ⎟⎜ ⎟′⊗⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
I 1 0
C I 1 0CC 0
0 I 1, where
( )I n Nm n N NM nm× −
⎛ ⎞′ ′= ⊗⎜ ⎟⎝ ⎠
C I 1 0 and
( )
( ) ( )
( ) ( )2
N n Nm N n NN M m
II N n nNm N N M mN Nm N n
−− × −
− ×−× −
′⎛ ⊗ ⎞⎛ ⎞⎜ ⎟⎜ ⎟′ = ⎜ ⎟⎜ ⎟′⊗⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
I 1 0C 0
0 I 1. The vector I I
II II
⎛ ⎞⎛ ⎞′= ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠
Y YC
Y Y has dimension
2 1N × with the first n random variables consisting of totals of the sample SSUs for each PSU,
namely 1
N
iI is s sIs
Y U m Y=
= ∑ , where 1
1 sm
sI sjjs
Y Ym =
= ∑ , 1,...,i N= , and the second set of ( )N n N− +
random variables consisting of totals for the potential second stage sample for the remaining
units, and for the remaining SSUs for each PSU, namely ( )1
N
iII is s s sIIs
Y U M m Y=
= −∑ , 1,...,i N=
with 1
1 s
s
M
sII sjj ms s
Y YM m = +
=− ∑ .
Let us consider the corresponding model. First,
( )
( ) ( )
( ) ( )
n Nm I
I I N n Nm N n NN M m
II IIIIN N M mN Nm N n
−− × −
−× −
⎛ ⎞′⊗⎜ ⎟
⎛ ⎞⎛ ⎞ ′⊗⎛ ⎞⎜ ⎟′= =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎜ ⎟′⊗⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
I 1 XX X I 1 0CX XX
0 I 1
where
( )
( )
1
1 2
1 1
1
s
N
I n Nm nNm n ms
n n N
N N
m m m mN
=
⎛ ⎞⎛ ⎞′= ⊗ ⊗ ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞= ⊗⎜ ⎟⎝ ⎠
X I 1 1 1 1
1 1
and
C06ed16.doc 5/1/2006 3:32 PM 35
( ) ( )
( ) ( )
( )
( ) ( ) ( )
( ) ( )
( )( )
1
1
1
1 1
1 1
1 1
1
s
s s
s
N
N n mN n NmN n Nm sN n NN M m
II NN N M mN Nm N n N M mN N M m s
N
N n Nm Nm N n mN n s
N NN M m N M
N N
N N
N N
N
−−− =− × −
−× −−− =
− −− =
−
⎛ ⎞⎛ ⎞⎛ ⎞⊗ ⊕′⊗ ⎜ ⎟⎜ ⎟⎛ ⎞ ⎜ ⎟⎝ ⎠⎝ ⎠⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟′⊗ ⎛ ⎞⎛ ⎞⎜ ⎟ ⊗ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠
⎛ ⎞⎛ ⎞′⊗ ⊗ ⊗ ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠=
′⊗ ⊗
1 1 1I 1 0X
0 I 11 1 1
I 1 1 1 1 1
I 1 1 1 ( ) ( )
( ) ( )
( ) ( ) ( ) ( )( )
1
1 2
1 1 2 2
1
1
1
s s
N
N M mm s
N n NN n
N N N N
N
m m m mN
M m M m M m M mN
−− =
−−
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞⎛ ⎞⊗ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⊗⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟= ⎜ ⎟⎛ ⎞− ⊗ − − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
1 1
1 1
1 1
or ( ) ( )
( ) ( ) ( ) ( )( )
1 2
1 1 2 2
1
1
N n NN n
II
N N N N
m m m mN
M m M m M m M mN
−−
⎛ ⎞⎛ ⎞⊗⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟= ⎜ ⎟⎛ ⎞− ⊗ − − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
1 1X
1 1.
Next, I I
II II
⎛ ⎞⎛ ⎞′= ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠
Z ZC
Z Z where
( )
( )
( ) ( )
( ) ( )
( )
( )
1 1
1 1
2
1 1
s
s
s s
n N
mn Nm i s nNm N n Nn N NM nm
N NI N n Nm N n NN M m mN n Nm nN i n sII
N n nNm N NN N M mN Nm N nM mi s
= = × −× −
−− × −
− × = + =
− ×−× −
−= =
⎛ ⎞⊕⊕′⊗⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞ ⎛ ⎞′ ⎜ ⎟⊗⎛ ⎞ ⊕ ⊕′ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟′⊗⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⊕⊕⎝ ⎠ ⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠⎝ ⎠
1 0I 1 0
Z I 1 0 0 1CZ 0
0 I 11
. Then
( )( )
( )( )
( )( )
1 1
1
1 2
s
s
n N
I n Nm mi s nNm N n N
N
n Nm n ms nNm N n N
n N n N n Nm m m
= = × −
= × −
× −
⎛ ⎞′= ⊗ ⊕⊕⎜ ⎟⎝ ⎠⎛ ⎞⎛ ⎞′= ⊗ ⊗ ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎛ ⎞= ⊗⎜ ⎟⎝ ⎠
Z I 1 1 0
I 1 I 1 0
I 0
and
C06ed16.doc 5/1/2006 3:32 PM 36
( ) ( ) ( )
( ) ( )
( )
( )
( )
1 1
1 1
1 1
s
s
s s
n N
mi s nNm N n N
N n Nm N NN n NN M mN n nNmmII N n Nm nN i n s
N N M mN Nm N nN nNm N N
M mi s
NN n nNm
N nNm
= = × −
−− × −− ×
− × = + =−× −×
−= =
− ×
×
⎛ ⎞⊕⊕⎜ ⎟′ ⎜ ⎟⎛ ⊗ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⊕ ⊕= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟′⊗⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎜ ⎟⎜ ⎟⊕⊕⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎛ ⎞⎜ ⎟=⎜ ⎟⎝ ⎠
1 0I 1 00
0 1Z0 I 10
1
I0
0
( ) ( )
( ) ( )
( )
( )
( )( )
( ) ( ) ( )( )
1 1
1
1
1 2
1 1 2 2
s
s
s s
n N
mi s nNm N n N
Nn Nm N n NN M mN n mN n Nm nN s
N N M mN Nm N n N
N M ms
N n NN n nN
N N N
m m m
M m M m M m
= = × −
−− × −
−− × =−× −
−=
−− ×
⎛ ⎞⊕⊕⎜ ⎟⎜ ⎟′⎛ ⊗ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟ ⊗ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟′⊗ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎜ ⎟⎛ ⎞⎜ ⎟⊗ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠⎝ ⎠
⊗⎛ ⎞⎜=⎜ ⊗ − − −⎝
1 0
1 00 I 1
0 I 1
I 1
0 I
I⎟⎟⎠
,
or ( )( )( )
( ) ( ) ( )( )( )1 2
1 1 2 2
N n NN n nNII
N N N
m m m
M m M m M m
−− ×
⎛ ⎞⎛ ⎞⊗⎜ ⎟⎜ ⎟⎝ ⎠= ⎜ ⎟⎜ ⎟⊗ − − −⎝ ⎠
0 IZ
I.
We summarize the model that collapses random variables to sample and remainder totals.
The model is given by
( )I I I I
II II II II
µ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞= + + + ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
Y X Z EM B
βY X Z E
where ( )1 21
I n n Nm m m mN
⎛ ⎞= ⊗⎜ ⎟⎝ ⎠
X 1 1 ,
( ) ( )
( ) ( ) ( ) ( )( )
1 2
1 1 2 2
1
1
N n NN n
II
N N N N
m m m mN
M m M m M m M mN
−−
⎛ ⎞⎛ ⎞⊗⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟= ⎜ ⎟⎛ ⎞− ⊗ − − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
1 1X
1 1,
C06ed16.doc 5/1/2006 3:32 PM 37
( )( )1 2I n N n N n N
m m m× −
⎛ ⎞= ⊗⎜ ⎟⎝ ⎠
Z I 0 ,
( )( )( )
( ) ( ) ( )( )( )1 2
1 1 2 2
N n NN n nNII
N N N
m m m
M m M m M m
−− ×
⎛ ⎞⎛ ⎞⊗⎜ ⎟⎜ ⎟⎝ ⎠= ⎜ ⎟⎜ ⎟⊗ − − −⎝ ⎠
0 IZ
I,
and I I
II II
⎛ ⎞⎛ ⎞′= ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠
E EC
E E.
Notice that
( )1 2
1
1
1
1
1
I n n N
N
n n s ss
N
n s ss
m m m mN
m mN
m mN
µ µ
µ β
µ β
=
=
⎛ ⎞ ⎛ ⎞⎛ ⎞= ⊗⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ ⎞= + ⎜ ⎟⎝ ⎠
⎛ ⎞= +⎜ ⎟⎝ ⎠
∑
∑
X 1 1β β
1 1
1
.
Now ( )1 1 1
1 1 1N N N
s s s s s ss s s
m m m mN N N
µ β µ β µ= = =
+ = + =∑ ∑ ∑ .
We have defined ssI s
mm
µ µ= and 1
1 Ns
I ss
mN m
µ µ=
= ∑ . As a result, ( )I n Imµ
µ⎛ ⎞
=⎜ ⎟⎝ ⎠
X 1β
.
Next,
( ) ( )
( ) ( ) ( ) ( )( )
( )
( ) ( )
1 2
1 1 2 2
1
1
1
1
N n NN n
II
N N N N
N n I
N
N N s s ss
m m m mN
M m M m M m M mN
m
M m M mN
µ µ
µ
µ β
−−
−
=
⎛ ⎞⎛ ⎞⊗⎜ ⎟⎜ ⎟⎝ ⎠⎛ ⎞ ⎛ ⎞⎜ ⎟=⎜ ⎟ ⎜ ⎟⎜ ⎟⎛ ⎞⎝ ⎠ ⎝ ⎠− ⊗ − − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞⎜ ⎟
= ⎜ ⎟⎛ ⎞− + −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∑
1 1X
β β1 1
1
1 1
.
C06ed16.doc 5/1/2006 3:32 PM 38
Notice that ( ) ( ) ( )1 1
1 1N N
s s s s s ss s
M m M m M mN N
µ β µ= =
⎛ ⎞− + − = −⎜ ⎟⎝ ⎠∑ ∑ . We have defined
s ssII s
M mM m
µ µ−=
− and
1
1 Ns s
II ss
M mN M m
µ µ=
−=
−∑ so that ( )
( )N n I
IIN II
m
M m
µµµ
−⎛ ⎞⎛ ⎞ ⎜ ⎟=⎜ ⎟ −⎜ ⎟⎝ ⎠ ⎝ ⎠
1X
1β. Using these
results, ( )
n n
I IN n N n
II IIN N
mm
M m
µ µµ
− −
⎛ ⎞⎜ ⎟⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎜ ⎟−⎝ ⎠⎝ ⎠
1 0X 1 0
βX 0 1. Let us define
( )
n n
IN n N n
IIN N
mm
M m− −
⎛ ⎞⎜ ⎟⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎜ ⎟−⎝ ⎠⎝ ⎠
1 0X 1 0X 0 1
where ( )I n nm=X 1 0 and ( )N n N n
IIN N
mM m
− −⎛ ⎞= ⎜ ⎟−⎝ ⎠
1 0X
0 1.
Then I II
II IIII
µ µµ
⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠
X XβX X
We can simplify ( ) ( )( )
( )1 2I n N n N n Nm m m
× −
⎛ ⎞+ = ⊗ +⎜ ⎟⎝ ⎠
Z M B I 0 M B in a similar
manner. Now ( )1 1
2 2
N N
⎛ ⎞+⎜ ⎟
+⎜ ⎟+ = ⎜ ⎟⎜ ⎟⎜ ⎟+⎝ ⎠
M BM B
M B
M B
where ( )( )i isM=M with ( )( )1is is isM U E Uξ µ= − and
( )( )i isB=B with ( )( )1is is is sB U E Uξ β= − . Then
( )( )
( ) ( )( )( )
( )( )
1 2 1 11 1
1 2 2 22 21 2
1 2
N
Nn N n N n N
N NN n n
m m m
m m mm m m
m m m
× −
⎛ ⎞+⎛ ⎞+ ⎜ ⎟⎜ ⎟ ⎜ ⎟++⎛ ⎞⎜ ⎟ ⎜ ⎟⊗ =⎜ ⎟⎜ ⎟⎝ ⎠ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟+⎝ ⎠ ⎜ ⎟+⎝ ⎠
M BM BM BM B
I 0
M B M B
where
C06ed16.doc 5/1/2006 3:32 PM 39
( )( ) ( )( )11 2 1 1 1 11
N
N s s s ss
m m m U E U mξ µ=
+ = −∑M B
so that
( )( )
( )( )
( )( )
( )( )
1
1
1
1 11
1 1
2 22 211 2
1
N
s s s ss
N
s s s ssn N n N n N
N N N
ns ns s ss
U E U m
U E U mm m m
U E U m
ξ
ξ
ξ
µ
µ
µ
=
=× −
=
⎛ ⎞−⎜ ⎟⎜ ⎟⎛ ⎞+⎜ ⎟⎜ ⎟ −+ ⎜ ⎟⎛ ⎞⎜ ⎟⊗ =⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟+ ⎜ ⎟⎝ ⎠⎜ ⎟−⎜ ⎟⎝ ⎠
∑
∑
∑
M BM B
I 0
M B
or ( )
( )( )
( )( )
( )( )
1
1
1
1 11
2 21
1
N
s s s ss
N
s s s ssI n
N
ns ns s ss
U E U m
U E U m
U E U m
ξ
ξ
ξ
µ
µ
µ
=
=
=
⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟
−⎜ ⎟+ = ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎜ ⎟⎝ ⎠
∑
∑
∑
Z M B I .
We have defined ( )( )1is is isM U E Uξ µ= − and ( )( )i isB=B . Also, ssI s
mm
µ µ= and
ssI
mrm
= . Let us define ( )( )11
N
iI is is sIs
M U E U rξ µ=
= −∑ and ( )( )11
N
iI is is sI ss
B U E U rξ β=
= −∑ . Then
( )( ) ( )1
1
N
is is s s iI iIs
U E U m m M Bξ µ=
− = +∑ . We represent ( )1 1
2 2
I I
I II I
NI NI
M BM B
M B
⎛ ⎞+⎜ ⎟
+⎜ ⎟+ = ⎜ ⎟⎜ ⎟⎜ ⎟+⎝ ⎠
M B . As a result,
( )( )
1 1
2 2
I I
I II n n N n
NI NI
M BM B
m
M B
× −
⎛ ⎞+⎜ ⎟
+⎛ ⎞⎜ ⎟+ = ⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟+⎝ ⎠
Z M B I 0 . Then ( )( )
( )I n I In N nm
× −
⎛ ⎞+ = +⎜ ⎟⎝ ⎠
Z M B I 0 M B or
C06ed16.doc 5/1/2006 3:32 PM 40
( )( ) ( )
( )( )
I I
I n n N n N n NII II
m× − − ×
⎛ ⎞+⎛ ⎞⎛ ⎞ ⎜ ⎟+ = ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ +⎝ ⎠
M BZ M B I 0 0
M B.
In a similar manner, let us define s ssII
M mrM m
−=
−, so that ( )( )1
1
N
iII is is sIIs
M U E U rξ µ=
= −∑
and ( )( )11
N
iII is is sII ss
B U E U rξ β=
= −∑ . Then
( ) ( )( )( )
( ) ( ) ( )( )( )( )
( ) ( )( )( )( ) ( )( )( )
( )( )
( )( )( )
( )( )
1
1
1
1
1
1 2
1 1 2 2
1 11
2 21
1
1 11
2 2
N n NN n nNII
N N N
N
s sn s n ssN
s sn s n ssN n
N
Ns Ns s ss
N
s s s s ss
s s sN
m m m
M m M m M m
U E U m
U E U m
U E U m
U E U M m
U E U M m
ξ
ξ
ξ
ξ
ξ
µ
µ
µ
µ
−− ×
+ +=
+ +=−
=
=
⎛ ⎞⎛ ⎞⊗⎜ ⎟⎜ ⎟⎝ ⎠+ = +⎜ ⎟⎜ ⎟⊗ − − −⎝ ⎠
⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟
−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎜ ⎟⎝ ⎠
=− −
− −
∑
∑
∑
∑
0 IZ M B M B
I
I
I( )
( )( )( )1
1
1
N
s ss
N
Ns Ns s s ss
U E U M mξ
µ
µ
=
=
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟− −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑
∑ .
. Let us define ( )1 1
2 2
II II
II IIII II
NII NII
M BM B
M B
⎛ ⎞+⎜ ⎟
+⎜ ⎟+ = ⎜ ⎟⎜ ⎟⎜ ⎟+⎝ ⎠
M B . Then
( ) ( )( )
( ) ( )N n I IN n n
II
N II II
m
M m
−− ×
⎛ ⎞⎛ ⎞ +⎜ ⎟⎜ ⎟⎝ ⎠+ = ⎜ ⎟⎜ ⎟− +⎝ ⎠
0 I M BZ M B
I M B or
C06ed16.doc 5/1/2006 3:32 PM 41
( ) ( ) ( )
( )( )( )
N n I IN n n N n NII
II IINN N
m
M m
−− × − ×
×
⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟ +⎜ ⎟⎝ ⎠ ⎜ ⎟+ = ⎜ ⎟⎜ ⎟+⎜ ⎟− ⎝ ⎠⎝ ⎠
0 I 0 M BZ M B
M B0 I.
Using these expressions,
( )( ) ( )
( ) ( )
( )
( )( )
n n N n N n N
I IIN nN n n N n N
II II II
NN N
m
m
M m
× − − ×
−− × − ×
×
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎛ ⎞+⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟+ =⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ +⎝ ⎠ ⎝ ⎠⎜ ⎟−⎜ ⎟⎜ ⎟
⎝ ⎠
I 0 0
M BZM B 0 I 0
Z M B0 I
. As a result, the collapsed
model is given by ( )I I I I
II II II II
µ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞= + + + ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
Y X Z EM B
βY X Z E or
( )
( ) ( )
( ) ( )
( )
( )( )
n n N n N n Nn n
I II I IN n N n
N nN n n N n NII II IIII IIN N
NN N
mm
m mM m
M m
µµ
× − − ×
− −−− × − ×
×
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠⎛ ⎞ ⎜ ⎟⎛ ⎞+⎜ ⎟⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎛ ⎞= + +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ +⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎜ ⎟− ⎝ ⎠⎜ ⎟⎝ ⎠⎝ ⎠ −⎜ ⎟⎝ ⎠
I 0 01 0
M BY E1 0 0 I 0Y EM B0 10 I
.
Let us consider the expected value of the terms in this model. The expected value is
given by
( )
( ) ( )
( ) ( )
( )
( )( )1 2 1 2 1 2
n n N n N n Nn n
I II I IN n N nN nN n n N n N
II II IIII IIN N
NN N
mm
mE E EmM m
M m
ξ ξ ξ ξ ξ ξ
µµ
× − − ×
− −−− × − ×
×
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠⎛ ⎞ ⎜ ⎟ ⎛ ⎞+⎜ ⎟⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟⎛ ⎞ ⎜ ⎟= + +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ +⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎜ ⎟− ⎝ ⎠⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ −⎜ ⎟⎜ ⎟⎝ ⎠
I 0 01 0
M BY E1 0 0 I 0Y EM B0 1
0 I
. Now ( )( )
( )( )
1 2
1 2
1 2
I I I I
II II II II
EE
E
ξ ξ
ξ ξ
ξ ξ
⎛ ⎞ ⎛ ⎞+ +⎜ ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠
M B M B
M B M B where ( )
( )( )
( )
1 2
1 2
1 2
1 2
1 1
2 2
I I
I II I
NI NI
E M B
E M BE
E M B
ξ ξ
ξ ξξ ξ
ξ ξ
⎛ ⎞+⎜ ⎟⎜ ⎟+
+ = ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+⎝ ⎠
M B and
C06ed16.doc 5/1/2006 3:32 PM 42
where ( ) ( )( )1 2 1 2 11
0N
ssI sI is is s
s
mE M B E U E Umξ ξ ξ ξ ξ µ
=
⎛ ⎞+ = − =⎜ ⎟⎝ ⎠∑ . Similarly, ( )
1 20sII sIIE M Bξ ξ + = .
In addition, 1 2 2
IN
II
Eξ ξ
⎛ ⎞=⎜ ⎟
⎝ ⎠
E0
E. As a result,
( )1 2
n n
I IN n N n
II IIN N
mmE
M mξ ξ
µµ
− −
⎛ ⎞⎜ ⎟⎛ ⎞ ⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟−⎝ ⎠⎝ ⎠
1 0Y 1 0Y 0 1
or
1 2
I II
II IIII
Eξ ξ
µµ
⎛ ⎞⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎝ ⎠
Y XY X
where ( )I n nm=X 1 0 and ( )N n N n
IIN N
mM m
− −⎛ ⎞= ⎜ ⎟−⎝ ⎠
1 0X
0 1.
The unbiased constraint for a linear predictor in the expanded model is given by
I iI I iII II⎡ ⎤′ ′ ′− + =⎣ ⎦L X g X g X 0 . We construct the constraint for the model collapsed to simple totals
next. First, let us consider the expression for i iiII I iiIII IIP ′ ′= +g Y g Y .
For the expanded model, ( )1 21 2 Ni i M M N Mc c c⎡ ⎤′ ′ ′ ′ ′= ⊗⎣ ⎦g e 1 1 1 and
( ) ( ) ( )
( )
** 1 1
* ** 1 1 * 1 1
s s
s s s
n N
ii N s M mn N n Nm i s n N n N M m
iII N N N N N
ii N s m ii N s M mi n s N n n i n s
e c
e c e c
−× − = = × − −
−= + = − × = + =
⎛ ⎞⎛ ⎞⎡ ⎤⎛ ⎞⎡ ⎤′ ′⊕ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠⎣ ⎦⎝ ⎠⎜ ⎟′ ′= ⎜ ⎟⎛ ⎞⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎜ ⎟⎡ ⎤′ ′ ′ ′⊕ ⊕ ⊕ ⊕⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎝ ⎠⎝ ⎠
0 1 1 0
g 11 1 0 1 1
. Now
( ) Ii i iI iII
II
P⎛ ⎞
′ ′ ′= = ⎜ ⎟⎝ ⎠
Yg Y g g
Y. We express this as
( )
( ) ( ) ( )( )1 2
1 2 1 2 1 2
* 1 2* 1
1 1 2 2 1 2 1 2
N
N N N
n
iI n ii m m N mi
i m m N m i m m N m iN m m N m
e c c c
e c c c e c c c e c c c
=
⎛ ⎞⎡ ⎤′ ′ ′ ′ ′= ⊕⎜ ⎟⎣ ⎦⎝ ⎠
′ ′ ′ ′ ′ ′ ′ ′ ′=
g 1 1 1 1
1 1 1 1 1 1 1 1 1
. Also,
C06ed16.doc 5/1/2006 3:32 PM 43
( )
( ) ( ) ((1 1 2 2
1 1 2 2 1 1 2 2 1 1 2 2
* * 1 2* 1 1 * 1
1 1 2 2 1 2 1 2
s s N N
N N N N
n N n
ii N s M m ii M m M m N M mi s i
i M m M m N M m i M m M m N M m iN M m M m
e c e c c c
e c c c e c c c e c c
− − − −= = =
− − − − − − − −
⎡ ⎤⎛ ⎞ ⎡ ⎤⎡ ⎤⎡ ⎤′ ′ ′ ′ ′⊕ ⊕ = ⊕⎜ ⎟⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦⎝ ⎠⎣ ⎦
′ ′ ′ ′ ′ ′ ′ ′=
1 1 1 1 1
1 1 1 1 1 1 1 1
Notice that for only one of these terms will * 1iie = . Let us assume that * 1iie = when * 2i = .
Then, ( )
( )( )1 2
1 2
* 1 2* 1
1 2
N
N
n
iI n ii m m N mi
Nm m m N m Nm
e c c c
c c c
=
⎛ ⎞⎡ ⎤′ ′ ′ ′ ′= ⊕⎜ ⎟⎣ ⎦⎝ ⎠
′ ′ ′ ′ ′=
g 1 1 1 1
0 1 1 1 0
and
( ) ( ) ( )
( )
** 1 1
* ** 1 1 * 1 1
s s
s s s
n N
ii N s M mn N n Nm i s n N n N M m
iII N N N N N
ii N s m ii N s M mi n s N n n i n s
N n Nm N
e c
e c e c
−× − = = × − −
−= + = − × = + =
−
⎛ ⎞⎛ ⎞⎡ ⎤⎛ ⎞⎡ ⎤′ ′⊕ ⊕⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠⎣ ⎦⎝ ⎠⎜ ⎟′ ′= ⎜ ⎟⎛ ⎞⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎜ ⎟⎡ ⎤′ ′ ′ ′⊕ ⊕ ⊕ ⊕⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎝ ⎠⎝ ⎠
′ ′ ′= ⊗
0 1 1 0
g 11 1 0 1 1
0 0 0 ( )( ) ( )( )( )1 1 2 21 2 N Nm M m M m N M m Nm N n N M mc c c− − − − −′ ′ ′ ′ ′ ′⊗1 1 1 0 0 0
.
Since we are considering simple totals, the value of 1sc = and hence
( )( )1 2 NiI Nm m m m Nm′ ′ ′ ′ ′ ′=g 0 1 1 1 0 , while
( )( ) ( )( )( )1 1 2 2 N NiII N n Nm Nm M m M m M m Nm N n N M m− − − − − −′ ′ ′ ′ ′ ′ ′ ′ ′ ′= ⊗ ⊗g 0 0 0 1 1 1 0 0 0 .
In order to collapse the expanded random variables to sample and remainder totals, we
pre-multiplied by
( )
( )
( ) ( )
( ) ( )2
n Nm n N NM nm
I N n Nm N n NN M m
IIN n nNm N N M mN Nm N n
× −
−− × −
− ×−× −
′⊗⎛ ⎞⎜ ⎟⎜ ⎟′⎛ ⎞ ′⊗⎛ ⎞′ = =⎜ ⎟ ⎜ ⎟⎜ ⎟′⎝ ⎠ ⎜ ⎟⎜ ⎟′⊗⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
I 1 0
C I 1 0CC 0
0 I 1, where
( )I n Nm n N NM nm× −
⎛ ⎞′ ′= ⊗⎜ ⎟⎝ ⎠
C I 1 0 and ( )
( ) ( )
( ) ( )2
N n Nm N n NN M m
II N n nNm N N M mN Nm N n
−− × −
− ×−× −
′⎛ ⊗ ⎞⎛ ⎞⎜ ⎟⎜ ⎟′ = ⎜ ⎟⎜ ⎟′⊗⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
I 1 0C 0
0 I 1. The vector
C06ed16.doc 5/1/2006 3:32 PM 44
I I
II II
⎛ ⎞⎛ ⎞′= ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠
Y YC
Y Y. In this expression, [ ]I n Nm I′= ⊗Y I 1 Y , and
( ) ( )
( ) ( )
N n Nm N n NN M m
II IIN N M mN Nm N n
−− × −
−× −
′⊗⎛ ⎞⎜ ⎟= ⎜ ⎟′⊗⎜ ⎟⎝ ⎠
I 1 0Y Y
0 I 1. As a result,
( )
( ) ( )
( ) ( )
n Nm I
I N n Nm N n NN M m
II IIN N M mN Nm N n
−− × −
−× −
⎛ ⎞′⊗⎜ ⎟
⎛ ⎞ ′⊗⎛ ⎞⎜ ⎟=⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎜ ⎟′⊗⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
I 1 YY I 1 0Y Y
0 I 1.
Notice that ( )( )n Nm n Nm nNm′′ ′⊗ ⊗ =I 1 I 1 I and that
( ) ( )( ) ( )1 1
n Nm n Nm n Nm n Nm n NmNm
− ⎛ ⎞⎡ ⎤′ ′′ ′ ′ ′⊗ ⊗ ⊗ ⊗ = ⊗⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠I 1 I 1 I 1 I 1 I J . In addition, notice that
1nNm n Nm n NmNm
⎛ ⎞− ⊗ = ⊗⎜ ⎟⎝ ⎠
I I J I P so that 1nNm n Nm n NmNm
⎛ ⎞= ⊗ + ⊗⎜ ⎟⎝ ⎠
I I J I P or equivalently that
( )( )1nNm n Nm n Nm n NmNm
′= ⊗ ⊗ + ⊗I I 1 I 1 I P . Similarly,
( ) ( ) ( )1N n Nm N n Nm N n NmN n Nm Nm − − −−
′= ⊗ ⊗ + ⊗I I 1 I 1 I P and
( ) ( ) ( )( ) ( )( ) ( )1
N N NNN M m N M m N M m N M mN M m− − − −′= ⊗ ⊗ + ⊗
−I I 1 I 1 I P . Using these expressions,
C06ed16.doc 5/1/2006 3:32 PM 45
( )
( )
( ) ( ) ( )
( ) ( )
( ) ( )( )
( )
( ) ( )( ) ( )
( ) ( ) ( )( ) ( )( )
2
2
1
1
1
nNm n N NM nm
N n Nm N n NN M m
N n nNmNN M mN Nm N n
n Nm n Nm n N NM nm
N n Nm N n Nm N n NN M m
N n nNmN NN M m N M mN Nm N n
Nm
Nm
N M m
× −
− − × −
− ×−× −
× −
− −− × −
− ×− −× −
⎛ ⎞⎜ ⎟⎜ ⎟⎛ ⎞ =⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞′⊗ ⊗⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞′⊗ ⊗⎜ ⎟⎜
⎜ ⎟⎜⎜ ⎟⎜ ′⊗ ⊗⎜ ⎟⎜ −⎜ ⎟⎜ ⎝ ⎠⎝ ⎠
I 0
I 00
0 I
I 1 I 1 0
I 1 I 1 0
00 I 1 I 1
( )
( )
( ) ( )
( ) ( )
( )( )
( )
( )( )
( ) ( ) ( ) ( )( )
( )( )
( )
2
2
2
1
1
1
n Nm n N NM nm
N n Nm N n NN M m
N n nNm N N M mN Nm N n
n Nm n NmnNm N n N NM nm n
N n Nm N n Nm N
N nn N NM nmN N M mN n NN M m
Nm
Nm
N M m
× −
−− × −
− ×−× −
× − − ×
− − ×
− ×× −−− × −
⎟ +⎟⎟⎟⎟
⊗⎛ ⎞⎜ ⎟⎜ ⎟⊗⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⊗⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞⊗ ′⊗⎜ ⎟⎜ ⎟⎜ ⎟⊗= ⎜ ⎟⎜ ⎟⎜ ⎟⊗⎜ ⎟−⎜ ⎟⎝ ⎠
I P 0
I P 00
0 I P
I 1 0 I 1 0
I 1 000
0 I 1
( )( ) ( )
( ) ( )( )
( )
( )
( ) ( )
( ) ( )2
N n Nm NN M m N n
nNmN N M mN N n Nm
n Nm n N NM nm
N n Nm N n NN M m
N n nNm N N M mN Nm N n
−− × −
−× −
× −
−− × −
− ×−× −
⎛ ⎞⎜ ⎟⎜ ⎟′⊗⎜ ⎟⎜ ⎟
′⊗⎜ ⎟⎜ ⎟⎝ ⎠
⊗⎛ ⎞⎜ ⎟⎜ ⎟⊗⎛ ⎞+⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⊗⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
I 1 0
0 I 1
I P 0
I P 00
0 I P
. Since
( )
( )
( ) ( )
( ) ( )2
n Nm n N NM nm
I N n Nm N n NN M m
IIN n nNm N N M mN Nm N n
× −
−− × −
− ×−× −
′⊗⎛ ⎞⎜ ⎟⎜ ⎟′⎛ ⎞ ′⊗⎛ ⎞′ = =⎜ ⎟ ⎜ ⎟⎜ ⎟′⎝ ⎠ ⎜ ⎟⎜ ⎟′⊗⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
I 1 0
C I 1 0CC 0
0 I 1,
C06ed16.doc 5/1/2006 3:32 PM 46
( )
( )
( ) ( ) ( )
( ) ( )
( )( )
( )
( )( )
( ) ( ) ( ) ( )( )
( )
( )
2
2
2
1
1
1
nNm n N NM nm
N n Nm N n NN M m
N n nNmNN M mN Nm N n
n Nm nNm N n
N n Nm N n Nm N
n N NM nmN N M mN n NN M m
n Nm n N NM nm
N n
N n nNm
Nm
Nm
N M m
× −
− − × −
− ×−× −
× −
− − ×
× −−− × −
× −
−
− ×
⎛ ⎞⎜ ⎟⎜ ⎟⎛ ⎞ =⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞⊗⎜ ⎟⎜ ⎟⎜ ⎟⊗ ′= ⎜ ⎟⎜ ⎟⎜ ⎟⊗⎜ ⎟−⎜ ⎟⎝ ⎠
⊗
⊗+
I 0
I 00
0 I
I 1 0
I 1 0 C0
0 I 1
I P 0
I0
( ) ( )
( ) ( )
Nm N n NN M m
N N M mN Nm N n
− × −
−× −
⎛ ⎞⎜ ⎟⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⊗⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
P 0
0 I P
Inserting this expression into ( ) Ii i iI iII
II
P⎛ ⎞
′ ′ ′= = ⎜ ⎟⎝ ⎠
Yg Y g g
Y,
( )
( )( )
( )
( )( )
( ) ( ) ( ) ( )( )
( )
( )
( )
( ) ( )
( ) ( )
2
2
1
1
1
n Nm nNm N n
IN n Nm N n Nm Ni iI iIIII
n N NM nmN N M mN n NN M m
n Nm n N NM nm
N n Nm N n NN M miI iII
N n nNm N N M mN Nm N n
Nm
P Nm
N M m
× −
− − ×
× −−− × −
× −
−− × −
− ×−× −
⎛ ⎞⊗⎜ ⎟⎜ ⎟⎜ ⎟ ⎡ ⎤⎛ ⎞⊗′ ′ ′= ⎜ ⎟ ⎢ ⎥⎜ ⎟
⎢ ⎥⎜ ⎟ ⎝ ⎠⎣ ⎦⎜ ⎟⊗⎜ ⎟−⎜ ⎟⎝ ⎠⊗
⊗⎛ ⎞′ ′+ ⎜⎜ ⊗⎜⎝ ⎠
I 1 0
YI 1 0g g CY0
0 I 1
I P 0
I P 0g g0
0 I P
I
II
⎛ ⎞⎜ ⎟⎜ ⎟⎛ ⎞
⎜ ⎟⎜ ⎟⎟ ⎝ ⎠⎜ ⎟⎟⎟⎜ ⎟⎝ ⎠
YY
.
Now
C06ed16.doc 5/1/2006 3:32 PM 47
( )
( )( )
( )
( )( )
( ) ( ) ( ) ( )( )
( )( )
( )
( ) ( ) ( ) ( )( )
2
1
1
1
11
1
n Nm nNm N n
N n Nm N n Nm NiI iII
n N NM nmN N M mN n NN M m
N n Nm N n Nm N
iI n Nm iII
N N M mN n NN M m
Nm
Nm
N M m
NmNm
N M m
× −
− − ×
× −−− × −
− − ×
−− × −
⎛ ⎞⊗⎜ ⎟⎜ ⎟⎜ ⎟⊗′ ′ =⎜ ⎟⎜ ⎟⎜ ⎟⊗⎜ ⎟−⎜ ⎟⎝ ⎠
⎛ ⎞⎛ ⎞⊗⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟′ ′⊗⎜ ⎟⎜ ⎟⊗⎜ ⎟⎜ ⎟−⎜ ⎟⎝ ⎠⎝ ⎠
I 1 0
I 1 0g g0
0 I 1
I 1 0g I 1 g
0 I 1
We assume the value of 1sc = and we assume that * 1iie = when * 2i = . Hence
( )( )1 2 NiI Nm m m m Nm iI Nm′ ′ ′ ′ ′ ′ ′ ′= = ⊗g 0 1 1 1 0 e 1 , while
( )( ) ( )( )( )( ) ( )( )( )
1 1 2 2 N NiII N n Nm Nm M m M m M m Nm N n N M m
iII Nm iI iIIN M m N M m
− − − − − −
− −
′ ′ ′ ′ ′ ′ ′ ′ ′ ′= ⊗ ⊗
′ ′ ′ ′ ′ ′= ⊗ ⊗ ⊗
g 0 0 0 1 1 1 0 0 0
e 1 e 1 e 1,
or iI iI Nm′ ′ ′= ⊗g e 1 while ( )( )iII iII Nm i N M m−′ ′ ′ ′ ′= ⊗ ⊗g e 1 e 1 .
As a result,
( ) ( )( )( )1 2
1 1NiI n Nm Nm m m m Nm n Nm iINm Nm
′ ′ ′ ′ ′ ′ ′⊗ = ⊗ =g I 1 0 1 1 1 0 I 1 e , while
( )( )
( ) ( ) ( ) ( )( ) ( )
1
1
N n Nm N n Nm N
iII N n i
N N M mN n NN M m
Nm
N M m
− − ×
−
−− × −
⎛ ⎞⊗⎜ ⎟⎜ ⎟′ ′ ′=⎜ ⎟⊗⎜ ⎟−⎝ ⎠
I 1 0
g 0 e0 I 1
. Then
C06ed16.doc 5/1/2006 3:32 PM 48
( )
( )( )
( )
( )( )
( ) ( ) ( ) ( )( )( )( )
2
1
1
1
n Nm nNm N n
N n Nm N n Nm NiI iII iI N n i
n N NM nmN N M mN n NN M m
Nm
Nm
N M m
× −
− − × −
× −−− × −
⎛ ⎞⊗⎜ ⎟⎜ ⎟⎜ ⎟⊗′ ′ ′ ′ ′=⎜ ⎟⎜ ⎟⎜ ⎟⊗⎜ ⎟−⎜ ⎟⎝ ⎠
I 1 0
I 1 0g g e 0 e0
0 I 1
.
Notice in addition that
( )
( )
( )
( ) ( )
( ) ( )
( )( ) ( )( ) ( ) ( )
( ) ( )
( )
2
n Nmn N NM nm
N n NmN n NN M miI iII
N n nNm N N M mN Nm N n
N n NmN n NN M m
iI Nm n Nm iII Nm i N M mN N M mN Nm N n
iI n Nm Nm
× −
−− × −
− ×−× −
−− × −
−−× −
⊗⎛ ⎞⎜ ⎟⎜ ⎟⊗⎛ ⎞′ ′ =⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⊗⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎛ ⊗ ⎞⎛ ⎞⎜ ⎟⎜ ⎟′ ′ ′ ′ ′ ′= ⊗ ⊗ ⊗ ⊗ =⎜ ⎟⎜ ⎟⊗⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
′ ′ ′= ⊗
I P 0
I P 0g g0
0 I P
I P 0e 1 I P e 1 e 1
0 I P
e I 1 P e ( ) ( )( )( )( ) ( )( )( )( )2
iII N n Nm Nm i N N M m N M m
iI Nm iII Nm i NN MM mm
N
− − −
−−
′ ′ ′⊗ ⊗ =
′ ′ ′ ′ ′ ′= ⊗ ⊗ ⊗
′=
I 1 P e I 1 P
e 0 e 0 e 0
0
Also, I I
II II
⎛ ⎞⎛ ⎞′= ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠
Y YC
Y Y. As a result,
C06ed16.doc 5/1/2006 3:32 PM 49
( )( ) ( )
( )
( )
( ) ( )
( ) ( )2
n Nm n N NM nm
I IN n Nm N n NN M mi iI N n i iI iIIII II
N n nNm N N M mN Nm N n
P
× −
−− × −−
− ×−× −
⊗⎛ ⎞⎜ ⎟⎜ ⎟⎛ ⎞⎛ ⎞ ⊗⎛ ⎞′ ′ ′ ′ ′= + ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟⎜ ⎟⊗⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
I P 0
Y YI P 0e 0 e g gY Y0
0 I P. Let us
define ( ) ( )( )iI iiIII iI N n i−′ ′ ′ ′ ′=g g e 0 e . Then
( ) ( )
( )
( )
( ) ( )
( ) ( )2
n Nm n N NM nm
I IN n Nm N n NN M mi iI iiIII iI iIIII II
N n nNm N N M mN Nm N n
P
× −
−− × −
− ×−× −
⊗⎛ ⎞⎜ ⎟⎜ ⎟⎛ ⎞⎛ ⎞ ⊗⎛ ⎞′ ′ ′ ′= + ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟⎜ ⎟⊗⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
I P 0
Y YI P 0g g g gY Y0
0 I P.
We require the predictor of i iiII I iiIII IIP ′ ′= +g Y g Y to be a linear function of the sample
random variables, i IP ′= L Y , and to be unbiased. This implies that
( ) ( )( )1 2 1 2ˆ 0Ii i iiII iiIII
II
E P P Eξ ξ ξ ξ
⎛ ⎞′ ′ ′− = − − =⎜ ⎟
⎝ ⎠
YL g g
Y. Since
1 2
I II
II IIII
Eξ ξ
µµ
⎛ ⎞⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎝ ⎠
Y XY X
where
( )I n nm=X 1 0 and ( )N n N n
IIN N
mM m
− −⎛ ⎞= ⎜ ⎟−⎝ ⎠
1 0X
0 1. This implies that
I iiII I iiIII II⎡ ⎤′ ′ ′− + =⎣ ⎦L X g X g X 0 . Now ( )0iiII I m′ =g X and
( ) ( ) ( )( )0N n N niiIII II N n i
N N
mM m
M m− −
−
⎛ ⎞′ ′ ′= = −⎜ ⎟−⎝ ⎠
1 0g X 0 e
0 1. As a result,
( )( )iiII I iiIII II m M m⎡ ⎤′ ′+ = −⎣ ⎦g X g X . This implies that ( ) ( )( )n nm m M m′ − − =L 1 0 0 . Thus,
without further restrictions, it is not possible to develop an unbiased linear predictor using
sample and remainder totals.
If second stage sampling is probability proportional to size, we can specify an unbiased
constraint for a linear predictor. To see this, notice that with PPS sampling, setting ss
MrM
= ,
C06ed16.doc 5/1/2006 3:32 PM 50
1
1 N
R I II s ss
rN
µ µ µ µ=
= = = ∑ . Then 1 2
11
I II IR
II IIII II
Eξ ξ
µµ
µ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞
= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
Y X XY X X
where 2I nm=X 1 1 and
( )2N n
IIN
m
M m−⎛ ⎞
= ⎜ ⎟⎜ ⎟−⎝ ⎠
1X 1
1. The unbiased constraint is given by nm M′ − =L 1 0 , or 1
n f′ =L 1
where mfM
= .
Collapsing to Weighted Totals
In order to collapse the expanded random variables to weighted sample and remainder
totals, we transform sty to * stst
s
yyM
= . Apart from this change, everything else is the same.
Notice that *
1 1
s sM Mst
s stt ts
y yM
µ= =
= =∑ ∑ so that the total corresponds to the PSU mean using transformed
data; similarly, *
*
1
1sMst
s st s s
yM M
µ µ=
= =∑ . Also, 1 2
* *
* *I II
II IIII
Eξ ξ
µµ
⎛ ⎞⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎝ ⎠
Y XY X
where
*
1
1 1Ns
I ss s
mN m M
µ µ=
⎛ ⎞= ⎜ ⎟
⎝ ⎠∑ and *
1
1 1Ns s
II ss s
M mN M m M
µ µ=
⎛ ⎞−⎛ ⎞= ⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠∑ . As a result,
( ) ( )
*1
*
1
1
1 1
N
s ssI
NII
s ss
fNm
fN M m
µµµ µ
=
=
⎛ ⎞⎜ ⎟⎛ ⎞ ⎜ ⎟=⎜ ⎟ ⎜ ⎟⎝ ⎠ −⎜ ⎟⎜ ⎟−⎝ ⎠
∑
∑. All of the modeling is identical, with the exception that
parameters are constructed using *sty instead of sty . The unbiased constraint is satisfied when
C06ed16.doc 5/1/2006 3:32 PM 51
second stage sampling is PPS. This follows since with PPS sampling,
( )
*
*1 1
11 1 11 1
N NI
s ss sII
fm
f N M NM m
µµ µ
µ = =
⎛ ⎞⎜ ⎟⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟−⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠ ⎜ ⎟−⎝ ⎠
∑ ∑ .
