A Class of Statistics with Asymptotically Normal Distribution

Let X 1 …, X n be n independent random vectors, X v = , and Φ(x 1 …, x m ) a function of m(≤n) vectors . A statistic of the form , where the sum ∑″ is extended over all permutations (α1 …, α m ) of different integers, 1 α≤ (αi≤ n, is called a U-statistic. If X 1, …, X n have the same (cumulative) distribution function (d.f.) F(x), U is an unbiased estimate of the population characteristic θ(F) = f … f Φ(x 1,…, x m ) dF(x 1) … dF(x m ). θ(F) is called a regular functional of the d.f. F(x). Certain optimal properties of U-statistics as unbiased estimates of regular functionals have been established by Halmos [9] (cf. Section 4)

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