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Assume Problem 2 finish,do Problem 4 onlySuppose that X = (Xi, X2, . . . , Xn) and Y = (y,Y2, . . . ,Yn) are random samples from continuous distributions F and G, respectively. Wilcoxons two-sample test statistic W = W(X,Y) is defined to be Σ-ngi Ri where Ri is the rank of in the combined sample.

2. Show that W can be written as where is the number of pairs (X,Y) with Xiくý, In other words Tn ΣΣΊ,) where I,,-10 otherwise U Hint: Let Y), Y2),..., Y(m) be the order statistics for the y-sample. Then U is the number of pairs (Xi,y(j)〉 with XìくY(). For fixed j, the number of Xi with Xi 〈 y(j) is just the Yank ofYo minus the number of Y k with Y、〈y(j), that is Řj-j. Ņow sum over j4. Explain why the identity W -Um(m1)/2 in Questions 2, shows that the value of Δ which minimises W(X, Y is given by

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