Definition
The term U-statistic, due to Hoeffding (1948), is defined as follows.
Let be a real-valued or complex-valued function of variables. For each the associated U-statistic is equal to the average over ordered samples of size of the sample values . In other words, the average being taken over distinct ordered samples of size taken from . Each U-statistic 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 ƒn(xφ) is exactly equal to the population value ƒN(x).
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