U-statistic

In statistical theory, a U-statistic is a class of statistics defined as the average over the application of a given function applied to all tuples of a fixed size. The letter "U" stands for unbiased. In elementary statistics, U-statistics arise naturally in producing minimum-variance unbiased estimators.

The theory of U-statistics allows a

Each U-statistic ${\displaystyle f_{n}}$ is necessarily a symmetric function.

U-statistics are very natural in statistical work, particularly in Hoeffding's context of

Fisher's k-statistics and Tukey's polykays are examples of homogeneous polynomial U-statistics (Fisher, 1929; Tukey, 1950).

For a simple random sample φ of size n taken from a population of size N, the U-statistic has the property that the average over sample values ƒ_n(xφ) is exactly equal to the population value ƒ_N(x).^{[clarification needed]}

Examples

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

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

If ${\displaystyle f(x_{1},x_{2})=(x_{1}-x_{2})^{2}/2}$, the U-statistic is the

${\displaystyle f_{n}(x)=\sum (x_{i}-{\bar {x}}_{n})^{2}/(n-1)}$

${\displaystyle n-1}$

${\displaystyle n\geq 2}$

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

The following case highlights an important point. If ${\displaystyle f(x_{1},x_{2},x_{3})}$ is the median of three values, ${\displaystyle f_{n}(x_{1},\ldots ,x_{n})}$ is not the median of ${\displaystyle n}$ values. However, it is a minimum variance unbiased estimate of the expected value of the median of three values, not the median of the population. Similar estimates play a central role where the parameters of a family of

See also

• V-statistic

Notes