101
and applying Corollary 1.7.1 in Serfling ( 1980 pg 25), it
f o1lows
'vVo ( o yV?
A U =
( n X i ) 2 ( 3 )' 2 (U1>ni- V2 )
(nAj)1^ (3)V2 (U2>n V2 )
(nX2)V2 (U3jnc)
(nX2)1/2 ( 3 )X/2 (U4 jn V2 )
> n( o, j:T)
where
I* T A|-uA'
1
1 2 P*
0
0
1 2P*
1
0
0
0
0
1
(3/4)1/2
0
0
- ( 3 / A )X/2
Recalling, that X^= lim ^ the proof of Theorem 4.2.1
n
follows.
The next step in proving the asymptotic distribution of
Wln tl 1 ^in will be to show that the estimators and M2
do not affect the asymptotic distribution of T. Theorem
4.2.2, states results about U-statistics with estimated
parameters (Randles, 1982). This theorem condenses
Conditions 2.2, 2.3, 2.9A, Lemma 2.6 and Theorem 2.13 from
Randles (1982) into the statement of Theorem 4.2.2.