Dependent Observations
(I) Quasi-likelihood estimating equations
Suppose now that
positive-definite
V Y    2V   , where V  
n n
is a symmetric
matrix of known functions
Vij   ,
no
longer diagonal. The quasi-score function is
U   
where
D  Dir n p
and
1

2
D tV 1  y    ,
Dir 
following properties,
EU    0, CovU   
 i
 r
. The score function has the
 U  
t 1
D
V
D

i


E

   .
2


1
Under suitable limiting conditions, the root
equation
̂ of the estimating
 
ˆ tVˆ 1  y  
ˆ 0 ,
U ˆ  D
 and asymptotically normally
is approximately unbiased for
distributed with limiting variance



1
Cov ˆ   2 DtV 1D  i1 .
Note:
Block-diagonal covariance matrices arise most commonly in
longitudinal studies, in which repeat measurements made on the
same subject are usually positively correlated. A well-known
application is the generalized estimating equation proposed by Liang
and Zeger (1986).
1
(II) Quasi-likelihood function
The quasi-score function for dependent observations is different from
the one for independent observations in that
U r   U s  
for dependent observatio ns 

 s
 r
 2Q  U r   U s    2Q 
for indpendent observatio ns 



 r  s
 s
 r
 s  r
The line integral
Q , y , t s  
t  s1  
1

2

t s0
t
1

y  t 1n V t nn dt n1 s  ,
 y
along a smooth path t s  in
Rn
from t s0   y
t s0    is path independent if and only if
 2Q , y, t s   2Q , y, t s 

 r  s
 s  r
where y   y1
y2
,
yn  .
t

Note:
Let
 Px, y 
F
, t s  


Q
x
,
y


Fdt s    Pxs , ys x ' s   Qxs , ys y ' s ds .
t b 
b
t a
a
 
 xs 
 y s  . Then,


For example,
 2 xy 
F 2
, t s  
2
x

y


 xs   s 
 y s    s 2 , 1  s  3 .

  
2
to
Then,
t 3 
 
t1

 



F  dt s    2s 3 1  s 2  s 4  2s ds   4s 3  2s 5 ds
3
1

3
1
968
3
◆
''
If V    b  ,
is some function, then
  b '  ,   b '    , b 
1
Q , y, t s 
is path-independent. Thus, we
can construct
V
1
    AtjW j A j  A j
k
j 1
which will ensure the path-independent of
construct
the
,
Q , y, t s  . After we
V 1   , we can further construct the
quasi-likelihood function. Consider the straight-line path
t s   y    y s, 0  s  1,
so that
t 0  y
and
t 1   .
Provided the straight-line path
exists, the quasi-likelihood function is given by
t 1
Q , y    y     2

If

1






s
V
t
s
ds
0
 y   

1
V 1   is approximately linear in t over the straight-line path,
the above integral may be approximated by
Q , y  
1
t


y


V 1   y   
2
3
1
t



y


V 1  y  y   
2
6
3