(a)
Conditional likelihood
Let
   ,   , where 
and

is the parameter vector of interest
is a vector of nuisance parameters. The conditional
likelihood can be obtained as follows:
S .
1. Find the complete sufficient statistic
2. Construct the conditional log-likelihood


lc  log f Y |S 
where
f Y |S 
Y  Y1
,
is the conditional distribution of the response

Y2
occur. One is that for fixed
Yn 
t
given
 0 , S   0 
S   0   S 
other is that
S  . Two cases might
depends on
is independent of
 0 . The
0
.The
following examples illustrate the use of conditional likelihood.
Example 1:

Y1 , Y2 ,, Yn i.i.d
 N  ,  2

 f  y   





1
2 2




n

 n
2
n

yi    


1 
 exp  i 1

2
2 
2



2 


n

 n

2
  yi  
  yi
2
n

i

1


i

1
exp  



2 2
2
2 2


1






 S  , S 2
statistics for
interest and
n
 n
    Yi ,  Yi 2  are complete sufficient
i 1
 i 1

 ,  . Therefore, if
2

 2 is
the parameter of
is the nuisance parameter, the conditional density
n
function
Y1 , Y2 ,, Yn given S    Yi is
f Y |S  





i 1
fY  y 
f S t 
 n 2  n


y

n


y

i


i
2 

1
n

i

1


i

1

 exp 


2
2
2
2 


2


2

2 




2
 t  n  


1

 exp 


2 
2n 2 
 2n 

 n 2

2 
  yi
t
n
 
exp  i 1 2  2 
2

2


2


 





2
2


 
C   
2
2


t
t
n  

exp 


2

 2 2 2  
 2 n



1  n 2 t 2 
2

  exp 
y  
2  i
2

n 
i

1


C
 
C
 
 
C2
 
C2
 
2


 n



  yi 
1  n 2  i 1 

 exp 
  yi 
2 2  i 1
n




 n
2 
   yi  y  

 exp  i 1
2
2





2







n
 

2  
2 


1


2n 2  
1
Therefore, the conditional log-likelihood is
n

   

lc  2  log C  2
which only depends on
inference for
2
2.
  yi
i 1
 y
2 2
2
,
Thus, we can conduct statistical
based on the conditional log-likelihood. Note that
the sufficient statistic for

, S 
n
Y
i 1
i
, is independent of
 2.
Example 2:
Y1 ~ N 1 ,1, Y2 ~ N  2 ,1 are independent. Suppose
 
2
1
is the parameter of interest and
  2
is the
nuisance parameter. Then, the sufficient statistic for the nuisance
parameter given
  0
is
 12   22
2 

S   0   Y1   0Y2 ~ N 
,
1


0  .


1


The conditional density is
3
  y1  1 2   y2   2 2 
 1 

 exp 

2
 2 


2
 
12   22  
 
  y1   0 y2 

1
1
 
exp  

2 1   02
2 1   02




2
fY | S  




  y2   0 y1     0 1 2 
 c 0  exp 

2
2
1


0



where
c 0  

1   02 . Thus,
2

y2   0 y1     0 1 
lc  , 1 ,  0   
 log c 0  .
2

2 1  0

Thus, we can make statistical inference for

based on
lc  , 1, 0  . For example, to find the maximum conditional
likelihood estimate, we can solve the score function
 l  , 1 ,  0  
U   c



  0 
 1
 y  0 y1    0 1 1 
   2   1 2

2
2
1


0

  0 


1  y2  y1 
1  2
0
4

 ˆ 
y2
y1
◆
(intuitively, it is a sensible estimate)
General Conditional Likelihood Approach:
(I)
S   0   S  independent of  0 , the conditional
For
log-likelihood (which only depends on


) can be obtained,


.
lc    log fY | S   log  fY  y   log f S   y 
Then,
̂ c
lc   is the maximum conditional
maximizing
likelihood estimate. To estimate the variance of
̂ c , the conditional
Fisher’s information can be used,
I | S 
Note that both
̂ c
and
  2lc  
 E  t
.






I | S 
are, in general, different from those
derived from the full likelihood.
(II)
For
S   0  dependent on  0 , the conditional log-likelihood
(which only depends on

 ,  , 0 ) can be obtained,


.
lc  ,  ,  0   log fY | S   0   log  fY  y   log f S   0   y 
Then,
̂ c
is the solution of
 l  ,  ,  0  
U   c
0

.



 0  ,   ˆ  
5
The asymptotic variance of
̂ c
is the inverse of

  2lc   
 E  t






  0  ,   ˆ   .

Note:
lc  ,  ,    l   ,  
is not the logarithm of a density and does
not ordinarily have the properties of a log-likelihood function.
6