103
Theorem 4 2.3 : Under HQ ,
(n fa [T* (y) T* (M.,M.)] 0
1 nl n ^ 1 2 J
"k tic
where Tn (y) is the statistic Tn which used y, while
tic
Tn is the statistic which use estimates of y,
and .
Proof:
The proof of Theorem 4.2.3. follows, if the conditions
of Theorem 4.2.2 hold. Although Theorem 4.2.2 has been
stated here in terms of one parameter, the theorem is valid
for a p-vector parameter (i.e., y). Thus, the unknown
parameter in this case is (y^,y2) which is being estimated
by (M^,M2). Next we need to show that the necessary
conditions hold.
Note, Condition 1 follows directly from the fact that
the kernel for Tn is an indicator function, that is
'P ( t) =
1 t> 0
0 t<0
and thus
|n|x2i-r2| IXn-Yil + |x2j-y2| | Xj j Y! | )
- Ixlj-u|>j
- 1'(|x2i~>J| |xli-p| + | x2 j
< 1.