The following case highlights an important point.
The starting point of the analysis of $ U $-
statistics is the Hoeffding decomposition of $ U $-
statistics, [a1]:
$$
U _ {n} ^ {m} ( \Phi ) = {\mathsf E} \Phi + \sum _ {c = r } ^ { m } \left ( \begin{array}{c}
m \\
c
\end{array}
\right ) U _ {n} ^ {c} ( g _ {c} ) ,
$$
where $ g _ {c} = g _ {c} ( x _ {1} \dots x _ {c} ) $,
$ r \leq c \leq m $,
are completely degenerate kernels: $ {\mathsf E} g _ {c} ( X _ {1} \dots X _ {c} ) = 0 $. For example, a single observation is itself an unbiased estimate of the mean and a pair of observations can be used to derive an unbiased estimate of the variance. There is a close connection with probability, since probability provides the mathematical language to describe uncertainty. Also, $ n ^ {- [ m ] } = {1 / {n ^ {[ m ] } } } $.
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The theory of U-statistics allows a minimum-variance unbiased estimator to be derived from each unbiased estimator of an estimable parameter (alternatively, statistical functional) for large classes of probability distributions. It is the science and art of making informed decisions in the face of uncertainty. The Department of Mathematics is thus cooperating with an university-wide Graduate Interdisciplinary Program in Statistics to provide support for statistics at the University of Arizona.
The contemporary development of the theory of $ U $-
statistics contains various generalizations: $ U $-
statistics with kernel taking values in a Hilbert or Banach space [a8], multi-sampling $ U $-
statistics, bootstrap and truncated $ U $-
statistics, weighted $ U $-
statistics, etc.
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Therefore, an $ U $-
statistic is an unbiased estimator of the functional $ \theta = {\mathsf E} \Phi $.
If
Full Report f
(
x
1
,
x
2
)
=
(
x
1
x
)
2
/
2
{\displaystyle f(x_{1},x_{2})=(x_{1}-x_{2})^{2}/2}
, the U-statistic is the sample variance
f
n
(
x
)
=
(
x
i
x
n
link
)
2
/
(
n
1
)
{\displaystyle f_{n}(x)=\sum (x_{i}-{\bar {x}}_{n})^{2}/(n-1)}
with divisor
n
1
{\displaystyle n-1}
, defined for
n
2
{\displaystyle n\geq 2}
. .