Suppose that X1, . . . , Xm form a random sample from a continuous Distribution for which the p.d.f. f (x) is unknown; Y1, . . . , Yn form an independent random sample from another continuous Distribution for which the p.d.f. g(x) also is unknown; and f (x) = g(x ˆ’ θ) for ˆ’ˆž H0: θ = θ0,
H1: θ = θ0.
Assume that no two of the observations are equal, and let Uθ0 denote the number of pairs (Xi, Yj) such that Xi ˆ’ Yj > θ0, where i = 1, . . . , m and j = 1, . . . , n. Show that for large values of m and n, the hypothesis H0 should not be rejected if and only if

where Φˆ’1 is the quantile function of the standard normal distribution.

mn(mn1 12 mnim+n 12