Ustatistics

U-statistics are a broad class of statistics used to estimate parameters that can be expressed as expectations of symmetric functions of samples. Given independent and identically distributed observations X1, …, Xn from a population with distribution F and a symmetric kernel h of k variables, the U-statistic of order k is defined as U_n = (1 / binomial(n, k)) sum h(X_{i1}, …, X_{ik}) over all 1 ≤ i1 < … < ik ≤ n. When k = 1, U_n is the sample mean, and for k = 2 many common statistics arise as U-statistics.

U-statistics are unbiased estimators of θ = E[h(X1, …, Xk)]. Under mild regularity conditions, they are often asymptotically efficient for this target parameter. A foundational result due to Hoeffding provides a decomposition of U_n into a sum of components (the Hoeffding decomposition), which clarifies variance contributions and the dependence on sample size. If the first projection φ1(x) = E[h(x, X2, …, Xk)] − θ is not identically zero, then sqrt(n)(U_n − θ) converges in distribution to Normal(0, σ^2), with σ^2 = Var(φ1(X)). If the kernel is degenerate (all lower projections vanish), the limit is non-normal and typically a weighted sum of independent chi-square variables.

Common examples and applications: Kendall's tau is a U-statistic of order 2 used for rank correlation. The Mann–Whitney statistic is another order-2 U-statistic used in nonparametric tests of stochastic dominance. U-statistics also provide unbiased estimators for a wide range of population functionals, including certain variances and measures of dispersion.

Computational notes: the number of kernel evaluations is binomial(n, k), which grows quickly with n and k; for practical use, one often relies on small k, exact formulas for special kernels, or Monte Carlo subsampling.