The U-statistic with a symmetric kernel φ(.) of degree m (≥ 1) is a symmetric, unbiased, and optimal nonparametric estimator of T(G). Defining T(Fn) as in (3.123), then (a) T(Fn) = Un for m = 1, but...

The U-statistic

with a symmetric kernel φ(.) of degree m (≥ 1) is a symmetric, unbiased, and optimal nonparametric estimator of T(G). Defining T(Fn) as in (3.123), then (a) T(Fn) = Un for m = 1, but (b) T(Fn) 6= Un for m ≥ 2. However, |T(Fn) − Un| = O(n −1 ) a.s. as n → ∞.