Chapter 3
U-STATISTICS

The tests met so far were each derived on their own terms: the sign statistic from the binomial, the signed-rank statistic from the symmetry of the parent, the rank-sum statistic from a counting argument. Each derivation was separate, and each variance was obtained by a calculation particular to it.

Hoeffding’s theory of \(U\)-statistics replaces all of that with a single construction. It begins from a question about estimation rather than testing — which parameters admit unbiased estimators, and what is the best such estimator — and the answer turns out to contain the mean, the variance, the Mann–Whitney statistic, Kendall’s tau and the signed-rank statistic as instances of one form. Their asymptotic normality then follows from one theorem rather than from several arguments, by way of the projection principle of Section 3.3: a \(U\)-statistic is shown to be asymptotically equivalent to a sum of independent terms, to which the ordinary central limit theorem applies.

This is the point at which the subject stops being a collection of tests and becomes a theory.