components of the form V
i 2
V
2
. Further, X
P
and Z
P
are known design matrices and y
P
is the N
x 1 vector of population y-values. The GLMM form :
» ¼
º «
¬ ª
» ¼
º «
¬ ª
» ¼
º «
¬ ª
» ¼
º «
¬ ª
e e
v Z
Z X
X y
y y
E
P
where the asterisk denotes non-sampled units. The vector of small area totals Y
i
is of the
form Ay + Cy with A = and C =
where = blockdiagA
T n
m i
i
1
1
T n
N m
i
i i
1
1
u m
i
A
1
1
, …, A
m
. We are interested in estimating a linear combination,
P = 1
T
E + m
T
v, of the regression
parameters
E and the realization of v, for specified vectors, l and m, of constants. For known
G, the BLUP best linear unbiased prediction estimator of P is given by Rao, 2003
H
P~ = tG, y = 1
T
E ~
+ m
T
v
~
= 1
T
E ~
+ m
T
GZ
T
V
-1
y - X
E ~
Model of indirect estimation,
z
T
i
ˆ
i T
E + b
i
v
i
+ e
i
, i = 1, …, m, is a special case of
GLMM with block diagonal covariance structure. Making the above substitutions in the general form for the BLUP estimator of
P
i
, we get the BLUP estimator of T
i
as:
H i
T ~
= z
i T
E ~
+ J
i
- z
i
Tˆ
i T
E ~
, where J
i
= V
v 2
b
i 2
\
i
+ V
v 2
b
i 2
, and E
~ =
E V ~
v 2
= »
» ¼
º «
« ¬
ª V
\ T
» »
¼ º
« «
¬ ª
V \
¦ ¦
m i
i v
i i
i m
i i
v i
T i
i
b b
1 2
2 1
1 2
2
ˆ
z z
z
4. State Space Models
Many sample surveys are repeated in time with partial replacement of the sample elements. For such repeated surveys considerable gain in efficiency can be achieved by
borrowing strength across both small areas and time. Their model consist of a sampling error model
T
it
ˆ
T
it
+ e
it
, t = 1, …, T; i = 1, …, m T
it
= z
it T
E
it
where the coefficients E
it
= E
it0
, E
it1
, …, E
itp T
are allowed to vary cross-sectionally and over time, and the sampling errors e
it
for each area i are assumed to be serially uncorrelated with mean 0 and variance
\
it
. The variation of E
it
over time is specified by the following model:
p j
v
itj ij
j t
i j
ij itj
,..., 1
, ,
1 ȕ
ȕ ȕ
ȕ
, 1
,
» ¼
º «
¬ ª
» ¼
º «
¬ ª
» ¼
º «
¬ ª
7
It is a special case of the general state-space model which may be expressed in the form
y
t
= Z
t
D
t
+ H
t
; E
H
t
= 0, E H
t
H
t T
= 6t
D
t
= H
t
D
t-1
+ A K
t
; E
K
t
= 0, E K
t
K
t T
= where
H
t
and K
t
are uncorrelated contemporaneously and over time. The first equation is known as the measurement equation, and the the second equation is known as the transition
equation. This model is a special case of the general linear mixed model but the state-space form permits updating of the estimates over time, using the Kalman filter equations, and
smoothing past estimates as new data becomes available, using an appropriate smoothing algoritm.
The vector D
t
is known as the state vector. Let
-1 t
Į~ be the BLUP estimator of D
t-1
based on all observed up to time t-1, so that
= H
-1 t
| t
Į~
-1 t
Į~ is the BLUP of D
t
at time t-1. Further,
P
t|t-1
= HP
t-1
H
T
+ A A
T
is the covariance matrix of the prediction errors
-1 t
| t
Į~ - D
t
, where
P
t-1
= E
1 -
t
Į~ - D
t-1
1 -
t
Į~ - D
t-1 T
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