U-statistics and Hoeffding decomposition

For n≥2 iid observations and a symmetric measurable order-two kernel h with $\mathbb E[h(X_1,X_2)^2]<\infty$, define θ as its expectation and average h over all unordered distinct pairs. Every term has expectation θ, so the U-statistic is unbiased. A V-statistic instead includes diagonal terms under a different normalization, and can have different finite-sample bias. The letter kernel here means a function of observations; it is not necessarily a nonnegative density-smoothing kernel.

Two pair terms sharing one observation can be correlated even though the original observations are independent. There are quadratically many pair terms, but their overlap often leaves a variance of order one over sample size rather than one over its square. Counting terms alone is therefore misleading. Symmetrization is useful when an original pair function is not symmetric, since averaging the two argument orders preserves its target expectation under iid sampling and creates the standard unordered-pair representation.

The first projection $h_1$ has expectation zero, and the canonical remainder satisfies $\mathbb E[h_2(X_1,X_2)\mid X_1]=0$ almost surely. Expanding each pair kernel and counting how often each observation appears gives the centered statistic as twice the average first projection plus the average remainder. The two components are orthogonal in L². Remainder terms sharing one argument also have zero covariance because of conditional centering, which gives an exact variance identity.

Let zeta1 be the variance of the first projection and zeta2 the variance of the canonical remainder. The first contribution is four zeta1 divided by n, while the second is two zeta2 divided by n times n minus one. These symbols refer to the centered canonical components; some sources use the same zeta notation for overlapping-kernel covariances instead. When zeta1 is positive, the ordinary iid central limit theorem applies to the first sum and the remainder is negligible at square-root-n scale.

If the first projection vanishes almost surely, the statistic is called degenerate at this distribution. Its variance is then of order one over n squared, provided the canonical variance is positive. A square-root-n normalization converges to zero rather than to a nontrivial normal distribution. The next-order limit is often a quadratic Gaussian expression, and depends on the kernel and distribution. Degeneracy is not a claim that the original observations have zero variance or that the statistic is identically constant.

This distinction matters in testing and resampling. A nondegenerate standard-error formula can collapse under a null where the first projection disappears, and a bootstrap designed for a linear projection may need modification. Moment conditions also matter: estimating variance with a squared-difference kernel needs a finite fourth moment for the square-integrable analysis used here. An algebraic unbiasedness identity can require fewer moments than a variance formula or limit theorem, so these levels of conclusion should not be bundled into one assumption-free statement.