A U-statistic averages a symmetric function of distinct sample observations over all unordered -tuples. For independent and identically distributed observations it is an unbiased estimator of . Omitting repeated indices can remove diagonal contributions in quadratic plug-in estimators.
For a symmetric degree- U-statistic kernel, the population functional is . Contaminating each of the identical distribution arguments and differentiating gives the displayed influence function. It equals times the first Hoeffding projection and gives leading asymptotic variance . If that projection vanishes, the usual root- normal limit degenerates and higher-order projections determine the limiting law.
A U-statistic kernel is an integrable function of a fixed number r of observations whose expectation is the parameter functional to be estimated. Exchangeability permits symmetrization without changing that expectation. This is a function of r sample values, rather than the one-displacement kernel for density estimation.
The symmetric kernel has expectation equal to the population variance for independent identically distributed observations with finite second moment. Its U-statistic reduces to the displayed unbiased sample variance, using .
Exchangeability of an independent identically distributed sample makes every permuted kernel have the same expectation. Averaging over permutations therefore leaves the target functional unchanged and gives a symmetric kernel. The U-statistic then averages this kernel over the unordered r-element sample subsets.
For an iid sample and symmetric square-integrable kernel of fixed order , put and . If , the displayed limit holds. The first Hoeffding projection is ; its orthogonal remainder is , giving the limit by the ordinary central limit theorem. Degenerate kernels require a different limiting theory.
The displayed sum is the first-order projection of a centred U-statistic onto sums of functions of individual observations. Its summands are independent and have mean zero. Subtracting it leaves a remainder with conditional expectation zero in each individual observation, a useful decomposition for variance and normal-limit calculations.
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A U-statistic is a type of statistic used in non-parametric statistical inference, particularly in estimating population parameters and testing hypotheses. It is designed to provide a way to estimate the value of a functional of a distribution based on a sample. U-statistics are particularly useful because they have desirable properties such as being asymptotically unbiased and having an asymptotic normal distribution. The general form of a U-statistic is constructed from a symmetric kernel function.