U-statistic

Definition

The term U-statistic, due to Hoeffding (1948), is defined as follows.

Let ${\displaystyle K}$ be either the real or complex numbers, and let ${\displaystyle f:(K^d{)}^r\to K}$ be a ${\displaystyle K}$-valued function of ${\displaystyle r}$ ${\displaystyle d}$-dimensional variables. For each ${\displaystyle n\ge r}$ the associated U-statistic ${\displaystyle f_n:(K^d{)}^n\to K}$ is defined to be the average of the values ${\displaystyle f(x_{i_1},\dots ,x_{i_r})}$ over the set ${\displaystyle I_{r,n}}$ of ${\displaystyle r}$-tuples of indices from ${\displaystyle \{1,2,\dots ,n\}}$ with distinct entries. Formally,

${\displaystyle f_n(x_1,\dots ,x_n)=\frac{1}{\prod _{i=0}^{r-1}(n-i)}\sum _{(i_1,\dots ,i_r)\in I_{r,n}}f(x_{i_1},\dots ,x_{i_r})}$.

In particular, if ${\displaystyle f}$ is symmetric the above is simplified to

${\displaystyle f_n(x_1,\dots ,x_n)=\frac{1}{(\genfrac{}{}{0ex}{}{n}{r})}\sum _{(i_1,\dots ,i_r)\in J_{r,n}}f(x_{i_1},\dots ,x_{i_r})}$,

where now ${\displaystyle J_{r,n}}$ denotes the subset of ${\displaystyle I_{r,n}}$ of increasing tuples.

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 independent and identically distributed random variables, or more generally for exchangeable sequences, such as in 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 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).

Examples

Some examples: If ${\displaystyle f(x)=x}$ the U-statistic ${\displaystyle f_n(x)={\overline{x}}_n=(x_1+\cdots +x_n)/n}$ is the sample mean.

If ${\displaystyle f(x_1,x_2)=|x_1-x_2|}$, the U-statistic is the mean pairwise deviation ${\displaystyle f_n(x_1,\dots ,x_n)=2/(n(n-1))\sum _{i>j}|x_i-x_j|}$, defined for ${\displaystyle n\ge 2}$.

If ${\displaystyle f(x_1,x_2)=(x_1-x_2{)}^2/2}$, the U-statistic is the sample variance ${\displaystyle f_n(x)=\sum (x_i-{\overline{x}}_n{)}^2/(n-1)}$ with divisor ${\displaystyle n-1}$, defined for ${\displaystyle n\ge 2}$.

The third ${\displaystyle k}$-statistic ${\displaystyle k_{3,n}(x)=\sum (x_i-{\overline{x}}_n{)}^{3}n/((n-1)(n-2))}$, the sample skewness defined for ${\displaystyle n\ge 3}$, 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, ${\displaystyle f_n(x_1,\dots ,x_n)}$ 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 probability distributions are being estimated by probability weighted moments or L-moments.