81
Lemma 3.4.3: Suppose that
U
n
where B is the set of all subsets of r integers chosen
without replacement from the set of integers {l,2,...,n} and
f(t ^ t2>..., t ) is some function symmetric in its r
arguments. This Un is a U-statistic of degree r with a
symmetric kernel f(). Let {nr} be an increasing sequence
of positive integers tending to infinity as r > and (Nr)
be a sequence of random variables taking on positive integer
2
values with probability one. If E{f(X^, X2.,X )} <
1/ o Nr P
lim VarCn^ U ) = r ? > 0, and 1 then
n 1 n
n+ r
r \ r2^)^ } = Mx) ,
lim P{(UN E(Un )) < N
r-)- r r
where $ ( ) represents the c.d.f. of a standard normal random
variable.
Proof: This is Lemma 3.3.3 in Popovich (1983). ^
One comment is needed about this result. The proof of this
lemma follows as a result of verifying that conditions
and C2 of Anscombe (1952) are valid and applying Theorem 1
of Anscombe (1952). Condition is valid under the null
hypothesis and the verification of condition C2 is contained
in the proof of Theorem 6 by Sproule (1974). This condition