Page 286 - DMTH404_STATISTICS

Statistics

Notes Thus

Sup L( |X,Y)

 Q 0


exp{ (m n)/2}

= m n

m n  m n 2

2 
2

[2 /(m n)] 2  (X  X)   (Y  Y) 

i
i
 1 1 
and
Sup L( |X,Y)

 Q 0


exp{ (m n)/2}
= m n
m
n
2 
2  
2


(2 /m) m /2 (2 /n) n /2   (X  X)   (Y  Y)  2
i
i
 1   1 
exp{ (m n)/2}


=
m n n
m
2 
2  
2
(2 /m) m /2 (2 /n) n /2   (X  X)   (Y  Y)  2


i
i
 1   1 
The likelihood ratio test is thus
Sup L( |X,Y)

 0
(X, Y) =
Sup L( |X,Y)


0
m n
 m 2   n 2  2
2
n /2  (X  X)   (Y  Y) 
m /2
 m   n   1 i   1 i 
=    



 m n   m n  m n m n
 2 2  2
 (X  X)   (Y  Y) 
i
i
 1 1 
Now
m n
 m 2   n 2  2
2
 (X  X)   (Y  Y) 
i
i
 1   1 

m n
 m 2 n 2  2
 (X  X)   (Y  Y) 
i
i
 1 1 
We have
 m  m /2  n  m /2
   


 m n   m n 
 (X, Y) = n /2 m /2
 (m 1)   (n 1) 


 1  f   1  (1/f) 


 (m 1)   (m 1) 
278 LOVELY PROFESSIONAL UNIVERSITY