Instatistical theory, aU-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.^{[citation needed]} In elementary statistics, U-statistics arise naturally in producingminimum-variance unbiased estimators.

The theory of U-statistics allows aminimum-variance unbiased estimator to be derived from eachunbiased estimator of anestimable parameter (alternatively,statisticalfunctional) for large classes ofprobability distributions.^[1][2] An estimable parameter is ameasurable function of the population'scumulative probability distribution: For example, for every probability distribution, the population median is an estimable parameter. The theory of U-statistics applies to general classes of probability distributions.

Many statistics originally derived for particular parametric families have been recognized as U-statistics for general distributions. Innon-parametric statistics, the theory of U-statistics is used to establish for statistical procedures (such as estimators and tests) and estimators relating to theasymptotic normality and to the variance (in finite samples) of such quantities.^[3] The theory has been used to study more general statistics as well asstochastic processes, such asrandom graphs.^[4][5][6]

Suppose that a problem involvesindependent and identically-distributed random variables and that estimation of a certain parameter is required. Suppose that a simple unbiased estimate can be constructed based on only a few observations: this defines the basic estimator based on a given number of observations. 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. The U-statistic based on this estimator is defined as the average (across all combinatorial selections of the given size from the full set of observations) of the basic estimator applied to the sub-samples.

Pranab K. Sen (1992) provides a review of the paper byWassily Hoeffding (1948), which introduced U-statistics and set out the theory relating to them, and in doing so Sen outlines the importance U-statistics have in statistical theory. Sen says,^[7] “The impact of Hoeffding (1948) is overwhelming at the present time and is very likely to continue in the years to come.” Note that the theory of U-statistics is not limited to^[8] the case ofindependent and identically-distributed random variables or to scalar random-variables.^[9]

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${\displaystyle I_{r,n}}$ ofincreasing tuples.

Each U-statistic${\displaystyle f_n}$ is necessarily asymmetric function.

U-statistics are very natural in statistical work, particularly in Hoeffding's context ofindependent and identically distributed random variables, or more generally forexchangeable sequences, such as insimple random sampling from a finite population, where the defining property is termed ‘inheritance on the average’.

Fisher'sk-statistics and Tukey'spolykays are examples ofhomogeneous polynomial U-statistics (Fisher, 1929; Tukey, 1950).

For a simple random sampleφ 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]}

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 thesample 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 sampleskewness 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 themedian 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 ofprobability distributions are being estimated by probability weighted moments orL-moments.

• V-statistic

• Borovskikh, Yu. V. (1996).U-statistics in Banach spaces. Utrecht: VSP. pp. xii+420.ISBN 90-6764-200-2.MR 1419498.
• Cox, D. R., Hinkley, D. V. (1974)Theoretical statistics. Chapman and Hall.ISBN 0-412-12420-3
• 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. (Partially reprinted in: Kotz, S., Johnson, N. L. (1992)Breakthroughs in Statistics, Vol I, pp 308–334. Springer-Verlag.ISBN 0-387-94037-5)
• Koroljuk, V. S.; Borovskich, Yu. V. (1994).Theory ofU-statistics. Mathematics and its Applications. Vol. 273 (Translated by P. V. Malyshev and D. V. Malyshev from the 1989 Russian original ed.). Dordrecht: Kluwer Academic Publishers Group. pp. x+552.ISBN 0-7923-2608-3.MR 1472486.
• Lee, A. J. (1990)U-Statistics: Theory and Practice. Marcel Dekker, New York. pp320ISBN 0-8247-8253-4
• Sen, P. K. (1992) Introduction to Hoeffding (1948) A Class of Statistics with Asymptotically Normal Distribution. In: Kotz, S., Johnson, N. L.Breakthroughs in Statistics, Vol I, pp 299–307. Springer-Verlag.ISBN 0-387-94037-5.
• Serfling, Robert J. (1980).Approximation theorems of mathematical statistics. New York: John Wiley and Sons.ISBN 0-471-02403-1.
• Tukey, J. W. (1950). "Some Sampling Simplified".Journal of the American Statistical Association.45 (252):501–519.doi:10.1080/01621459.1950.10501142.
• Halmos, P. (1946)."The Theory of Unbiased Estimation".Annals of Mathematical Statistics.1 (17):34–43.doi:10.1214/aoms/1177731020.

• U-statistics
• Estimation theory
• Nonparametric statistics
• Asymptotic theory (statistics)

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