Conditional Likelihood for Exponential Family:
Suppose that the log-likelihood for
   ,  
can be written in
the exponential family form
l  , y    t s  b  .
l  , y  has a decomposition of the form
Also, suppose
l  , y    t s1  t s2  b ,   .
Note:

is a linear

is arbitrary
The above decomposition can be achieved only if
function of

. The choice of nuisance parameter
and the inferences regarding
parameterization chosen for

should be unaffected by the
.
◆
The conditional likelihood of the data Y given
s2 is
l  | s2    t s1  b  , s2  ,
which is independent of the nuisance parameter and may be used for
inferences regarding

.
Example :
Y1 ~ P1 , Y2 ~ P 2 
are
independent.
Suppose
 2 

  log  2   log 1  is the parameter

 1
  log 
1
of interest and
1  log 1 
is the nuisance parameter. Then,
the log-likelihood is
l  , 1 

 log e   1   2  1y1  2y 2

 1   2   y1 log 1   y 2 log  2 

2 

  1 
1


  y1 log 1   y 2 log 1 

1 

 y 2 log 1   log  2 


 e 1 1  e   y1  y 2 1  y 2
 s1  s2 1  b , 1 

 s1  y2 , s2  y1  y2 , b , 1   e 1 1  e
Then, the conditional distribution of
Y1 , Y2 given s2  Y1  Y2  m

1 


B
m
,
is      . Thus,
1
2

l  | s 2  m 

1
 y1 log 
 
2
 1


2



y
log
2

 
2

 1





1
 y1 log 
 
2
 1


1



y
log
2

 
2

 1






1
 y 2 log 
 
2
 1

.


2



log

 
2

 1




1









1
2


1




  y1  y 2  log 
1
  y 2


1

exp









2

 1 


1


 s1  b   , s 2 
2


1




b

,
s


s
log
2
2
 1  exp    .
where


Note:
We can choose the other nuisance parameters, for example,
2  log  2 , 3  log 1   2 , 4  log 1   log  2  .
Thus, the log-likelihood is
l  , 2 
 1   2   y1 log 1   y 2 log  2 

 y1   y1  y 2 2  e 2 1  e

l  , 4 
 y  y2     y1  y2  
 1
2
4
2
 2e
4
2
 
cosh  
2
No matter what parameterization is chosen for the nuisance
parameter, the sufficient statistic
S 2  Y1  Y2
has the same form.
Example :
Y1 ~ B1, 1 , Y2 ~ B1, 2  are independent. Suppose
 2
1 



log

1
1


1 
2



  log 
interest and
 2
1 2
  log 
Then, the log-likelihood is
3


 is the parameter of



 is the nuisance parameter.

l  ,  

 2y 1   2 1 y
 y1 log  1   1  y1  log 1   1 
 y2 log  2   1  y2  log 1   2 
 log  1y1 1   1 
1 y1
2
2
 2
 1 


log
y

 y1 log 
2
1 
1  
2
1 


 log 1   1   log 1   2 





 2 
 1 


log
y

 y1 log 
1

1  
1  
2 
1 


 2 
  y1  y2  log 
  log 1   1   log 1   2 
1  
2 

 y1    y1  y2   log 1   1   log 1   2 
 s1  s2  b ,  
 s1  y1 , s2  y1  y2 ,
b ,    log 1   1   log 1   2  .
Then, the conditional likelihood can be obtained based on the
conditional distribution of
Y1 , Y2 given s2  Y1  Y2  m , which is
the hypergeometric distribution as
4
1   2 .