In statistical theory, a Ustatistic is a class of statistics that is especially important in estimation theory; the letter "U" stands for unbiased. In elementary statistics, Ustatistics arise naturally in producing minimumvariance unbiased estimators.
The theory of Ustatistics allows a minimumvariance unbiased estimator to be derived from each unbiased estimator of an estimable parameter (alternatively, statistical functional) for large classes of probability distributions.^{[1]}^{[2]} An estimable parameter is a measurable function of the population's cumulative probability distribution: For example, for every probability distribution, the population median is an estimable parameter. The theory of Ustatistics applies to general classes of probability distributions.
Many statistics originally derived for particular parametric families have been recognized as Ustatistics for general distributions. In nonparametric statistics, the theory of Ustatistics is used to establish for statistical procedures (such as estimators and tests) and estimators relating to the asymptotic normality and to the variance (in finite samples) of such quantities.^{[3]} The theory has been used to study more general statistics as well as stochastic processes, such as random graphs.^{[4]}^{[5]}^{[6]}
Suppose that a problem involves independent and identicallydistributed 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 Ustatistic 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 subsamples.
Sen (1992) provides a review of the paper by Wassily Hoeffding (1948), which introduced Ustatistics and set out the theory relating to them, and in doing so Sen outlines the importance Ustatistics 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 Ustatistics is not limited to^{[8]} the case of independent and identicallydistributed random variables or to scalar randomvariables.^{[9]}
Contents

Definition 1

Examples 2

See also 3

Notes 4

References 5
Definition
The term Ustatistic, due to Hoeffding (1948), is defined as follows.
Let f\colon R^r\to R be a realvalued or complexvalued function of r variables. For each n\ge r the associated Ustatistic 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) = \operatorname{ave} f(x_{\varphi(1)},\ldots, x_{\varphi(r)}), the average being taken over distinct ordered samples of size r taken from \{1,\ldots, n\}. Each Ustatistic f_n(x_1,\ldots, x_n) is necessarily a symmetric function.
Ustatistics are very natural in statistical work, particularly in Hoeffding's context of independent and identicallydistributed 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 kstatistics and Tukey's polykays are examples of homogeneous polynomial Ustatistics (Fisher, 1929; Tukey, 1950). For a simple random sample φ of size n taken from a population of size N, the Ustatistic 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 f(x) = x the Ustatistic 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 Ustatistic is the mean pairwise deviation f_n(x_1,\ldots, x_n) = \sum_{i\neq j} x_i  x_j / (n(n1)), defined for n\ge 2.
If f(x_1, x_2) = (x_1  x_2)^2/2, the Ustatistic is the sample variance f_n(x) = \sum(x_i  \bar x_n)^2/(n1) with divisor n1, defined for n\ge 2.
The third kstatistic k_{3,n}(x) = \sum(x_i  \bar x_n)^3 n/((n1)(n2)), the sample skewness defined for n\ge 3, is a Ustatistic.
The following case highlights an important point. If f(x_1, x_2, x_3) is the median of three values, f_n(x_1,\ldots, x_n) is not the median of n values. However, it is a minimum variance unbiased estimate of the expected value of the median of three values and in this application of the theory it is the population parameter defined as "the expected value of the median of three values" which is being estimated, 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 Lmoments.
See also
Notes

^ Cox & Hinkley (1974),p. 200, p. 258

^ Hoeffding (1948), between Eq's(4.3),(4.4)

^ Sen (1992)

^ Page 508 in

^ Pages 381–382 in

^ Page xii in

^ Sen (1992) p. 307

^ Sen (1992), p306

^ Borovskikh's last chapter discusses Ustatistics for exchangeable random elements taking values in a vector space (separable Banach space).
References

Cox, D.R., Hinkley, D.V. (1974) Theoretical statistics. Chapman and Hall. ISBN 0412124203

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. SpringerVerlag. ISBN 0387940375)

Lee, A.J. (1990) UStatistics: Theory and Practice. Marcel Dekker, New York. pp320 ISBN 0824782534

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. SpringerVerlag. ISBN 0387940375.
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