Theory of Statistics - George Mason University

402 5 Unbiased Point Estimation
5.2.2 Kernels and U-Statistics
Now consider the estimation of the expectation functional Θ(P) in equation
(5.32), given a random sample X1, . . ., Xn, where n ≥ m.
Clearly h(X1, . . ., Xm) is an unbiased estimator of θ = Θ(P), and so is
h(Xi1, . . ., Xim) for any m-tuple, 1 ≤ i1 < · · · < im ≤ n; hence, we have that
U = 1
n
h(Xi1, . . ., Xim) (5.35)
m all combinations
is unbiased for θ.
A statistic of this form is called a U-statistic. The U-statistic is a function
of all n items in the sample. The function h, which is called the kernel of the
U-statistic is a function of m arguments. We also refer to the order of the
kernel as the order of the U-statistic.
Examples
Example 5.15 rth raw moment: M ′ r(P) = E(Xr )
In the simplest U-statistic for r = 1, the kernel is of order 1 and h is the
identity, h(x) = x. This is just the sample mean. More generally, we have the
, yielding the first order
r th raw population moment by defining hr(xi) = x r i
U-statistic
U(X1, . . ., Xn) = 1
n
n
i=1
X r i ,
which is the r th sample moment.
(The notation hr will be used differently below.***)
Example 5.16 r th power of the mean: (E(X)) r
Another simple U-statistic with expectation (E(X)) r where the r th order kernel
is h(x1, . . ., xr) = x1 · · ·xr. The U-statistic
n
r
U(X1, . . ., Xn) = 1
has expectation (E(X)) r .
all combinations
Xi1 · · ·Xir
Example 5.17 Pr(X ≤ a): Θ(P) = E I]∞,a](X) = P(a)
Compare this with the quantile functional in equation (5.31), which cannot
be expressed as an expectation functional. The quantile problem is related to
an inverse problem in which the property of interest is the π; that is, given a
value a, estimate P(a). We can write an expectation functional and arrive at
the U-statistic
Theory of Statistics c○2000–2013 James E. Gentle