In Example 2.1, if the null hypothesis is true, \(F_{X}=F_{0}\). Thus, \(\mathcal{E}(X, Y)=2 E|X-Y|-E\left|X-X^{\prime}ight|-E\left|Y-Y^{\prime}ight|=0\). Show that for the \(U\)-statistic, \(E\left(\mathcal{E}_{n}^{*}ight)=0\). That is, show that the \(U\)-statistic given by (2.3) is unbiased for the energy distance \(\mathcal{E}(X, Y)\).

Example 2.1

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As an example of a statistical function, consider the sample variance. If E(X) = , then T(F)=o= f(-u)dFx (r) is the statistical functional. The corresponding statistical function substitutes dFn() == for dF, and we get the sample variance 62 = -1 (-T). 72 Consider the Cramr-von-Mises and Anderson-Darling goodness-of-fit statistics for testing Ho F = Fo vs H F Fo for some specified null distribution Fo. Both statistics are based on a functional of this form: T(F) = [(F(x) Fo(x)) w(r; Fo) dFo(2), where w(r; Fo) is a given weight function depending on Fo. When w(r, Fo) = 1, the statistical function is the well-known Cramr-von-Mises test statistic, substituting F for F. For the Anderson-Darling test, the weight function is [Fo(r)(1 - Fo(2))].