Theory of Statistics - George Mason University

5.2 U-Statistics 409
Theorem 5.4 (Hoeffding’s Theorem)
Let U be a U-statistic with m th order kernel h with E(h(X1, . . ., Xm) 2 ) < ∞.
Then
where
Proof.
MS2 p. 176.
Projections of U-Statistics
V(U) = 1
m
m n − m
k m − k
n
m
k=1
ζk
(5.52)
ζk = V(hk(X1, . . ., Xk)). (5.53)
One method of working out the asymptotic distribution of a U-statistic is by
use of projections
We first relate Theorem 1.65 on page 118 to the U-statistic,
Un = 1
n
h(Xi1, . . ., Xim).
m all combinations
Let Un be the projection of Un onto X1, . . ., Xn. (See Section 1.5.3 beginning
on page 115.) Recall, as in equation (5.48),
h1(x1) = E(h(X1, X2, . . ., Xm)|X1 = x1)
= E(h(x1, X2, . . ., Xm)).
and
˜h1 = h1 − E(h(X1, . . ., Xm)),
Then, starting with the definition of Un as a projection, we have
n
Un = E(Un) + (E(Un|Xi) − E(Un))
i=1
= E(Un) + m
n
˜h1(Xi).
n
i=1
This yields
V(Un) = m2
n ζ1,
where, in the notation of equation (5.53), ζ1 = V(h1(X1)).
Hence, by Hoeffding’s theorem (actually a corollary of it), and Theorem
1.65, we have
E((Un − Un) 2 ) ∈ O(n −2 ).
If ζ1 > 0, this yields
(Theorem 3.5(i) in MS2.)
√ n(Un − E(Un)) d → N(0, m 2 ζ1).
Theory of Statistics c○2000–2013 James E. Gentle