U-statistic

The term U-statistic, due to Hoeffding (1948), is defined as follows. Let $f\colon R^{r}\to R$ be a real-valued or complex-valued function of $r$ variables. For each $n\geq r$ the associated U-statistic $f_{n}\colon R^{n}\to R$ is equal to the average over ordered samples $\varphi (1),\ldots ,\varphi (r)$ of size $r$ of the sample values $f(x\varphi )$. In other words, $f_{n}(x_{1},\ldots ,x_{n})=\mathop {ave} f(x_{\varphi (1)},\ldots ,x_{\varphi (k)})$, the average being taken over distinct ordered samples of size $r$ taken from $\{1,\ldots ,n\}$. Each U-statistic $f_{n}(x_{1},\ldots ,x_{n})$ is necessarily a symmetric function.

U-statistics are very natural in statistical work, particularly in Hoeffding's context of independent and identically distributed random variables. They also arise naturally in the context of simple random sampling from a finite population, where the defining property is termed `inheritance on the average'. Fisher's $k$-statistics and Tukey's polykays are examples of homogeneous polynomial U-statistics (Fisher, 1929; Tukey, 1950). For a simple random sample $\varphi$ of size $n$ taken from a population of size $N$, the U-statistic has the property that the average over sample values $f_{n}(x\varphi )$ is exactly equal to the population value $f_{N}(x)$.

Some examples: If $f(x)=x$ the U-statistic $f_{n}(x)={\bar {x}}_{n}=(x_{1}+\cdots +x_{n})/n$ is the sample mean.

If $f(x_{1},x_{2})=|x_{1}-x_{2}|$, the U-statistic is the mean pairwise deviation $f_{n}(x_{1},\ldots ,x_{n})=\sum _{i\neq j}|x_{i}-x_{j}|/(n(n-1))$, defined for $n\geq 2$.

If $f(x_{1},x_{2})=(x_{1}-x_{2})^{2}/2$, the U-statistic is the sample variance $f_{n}(x)=\sum (x_{i}-{\bar {x}}_{n})^{2}/(n-1)$ with divisor $n-1$, defined for $n\geq 2$.

The third $k$-statistic $k_{3,n}(x)=\sum (x_{i}-{\bar {x}}_{n})^{3}n/((n-1)(n-2))$, the sample skewness defined for $n\geq 3$, is a U-statistic.

References

Fisher, R.A. (1929) Moments and product moments of sampling distributions. Proceedings of the London Mathematical Society, 2, 30:199-238.

Hoeffding, W. (1948) A class of statistics with asymptotically normal distributions. Annals of Statistics, 19:293-325.

Lee, A.J. (1990) U-Statistics: Theory and Practice. Marcewl Dekker, New York. pp320 ISBN 0824782534

Tukey, J.W. (1950) Some Sampling Simplified. J. Amer. Statist. Assoc. 45:501-519.