102
Theorem 4.2.2: Given the following three conditions
1)Assume there exists a B^>0 such that
< for every
Xj x and all y in some neighborhood of p where
h( ;t) denotes the kernal of the U-statistic Un.
2)Suppose there is a neighborhood of X, call it K(X)
and a constant B2>0 such that if yeK(X) and
D(y,d) is a sphere centered at y with radius d
satisfying D(y,d) c K(X) then
E [ Sup |h(X
y e D ( y d )
xr;T)|]
.,Xr;y') h(Xx
> y
(Condition 2.3)
3)Assume [h(X^,...,Xf;y)] has a zero differential
at y =y, that
and
where
a
2 = Var{E[h(X1,...,Xr;u)|xi]} > 0
(Condition 2 9A)
then
n/2 [U (i) U (y ) ] > 0
L n n J
Proof:
See Randles (1982).