Page 282 - DMTH404_STATISTICS

Statistics

Notes In order to obtain the test statistic, we use the likelihood ratio test. We have

2
 = { ( ,  ,  ) : – <  ,  < ,  > 0 }
2
1 2 1 2
 = {  =  =  (say),  ) : –  <  < ,  > 0}
2
2
0 1 2
We shall write  = ( ,  ,  )
2
l 2
We have

{sup L( |X, Y)
 Q 0
1  1  m n 2  
2
= Sup m n m n exp  2  (X   1 )   (Y   2 
) 

i
i



(2 ) 2 ( 2 ) 2  2  1 1 
Under H ,  =  =  and the maximum likelihood estimate of  is
0 1 2

mX nY
2
ˆ  = and of  is
m  n
1  m n mn 
2
2
2
ˆ  =  (X   1 )   (X   2 )  (X Y) 2 

1
i
m n  1 1 (m n) 


= u (say)
Sup L ( |X, Y) 1  1 


Thus = m n exp  (m n)u' 


 0  2u' 
(2pu') 2
m n

 1  2  (m  n) 
=   exp  


 2 u'  2 
Under H , the maximum likelihood estimates of  ,  and  are respectively
2
1 1 2
m n
2
 (X  X)   (Y Y) 2

i
ˆ 
ˆ   X, Y, 2 1 1  u(say)
ˆ 
1 2
m  n
and
{sup L( |X, Y)

 Q 0
m n

n
 1  2  m  
=   exp  


 2 u   2 
The likelihood ratio test is thus
{sup L( |X, Y)

 0
(X, Y) =
{sup L( |X, Y)


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