Theory of Statistics - George Mason University

408 5 Unbiased Point Estimation
Proof.
We first define an n th order expectation kernel to be associated with T; in
fact, it is the function itself:
The associated U-statistic is
h(Xi1, . . ., Xin) = T(Xi1, . . ., Xin).
U = 1
n!
T(Xi1, . . ., Xin)
Now, as in equation (5.44), we write
U = E
T(Xi1, . . ., Xin)|X(1), . . ., X(n) .
Hence,
C
E U 2 E
2
= E T |X(1), . . ., X(n)
≤ E E T 2
|X(1), . . ., X(n)
= E T 2 .
with equality if and only if E
T |X(1), . . ., X(n) is degenerate and equals T
with probability 1.
We will assume E(h(X1, . . ., Xm) 2 ) < ∞. We first introduce some additional
notation for convenience.
(The notation hr problem ***)
For k = 1, . . ., m, let
hk(x1, . . ., xk) = E(h(X1, . . ., Xm)|X1 = x1, . . ., Xk = xk)
= E(h(x1, . . ., xk, Xk+1, . . ., Xm)). (5.48)
We have hm = h and
hk(x1, . . ., xk) = E(hk+1(x1, . . ., xk, Xk+1, . . ., Xm)). (5.49)
Now, we define the centered versions of the h: for k = 1, . . ., m,
˜hk = hk − E(h(X1, . . ., Xm)), (5.50)
and let
˜h = ˜hm We see that the corresponding centered U-statistic is
U − E(U) = 1
˜h(Xi1, . . ., Xim) (5.51)
n
m
C
This notation is convenient in the demonstration that a sequence of adjusted
kernels forms a martingale (see Serfling (1980), page 177).
It is also a simple matter to work out the variance of the corresponding
U-statistic.
Theory of Statistics c○2000–2013 James E. Gentle