Consider testing the null hypothesis that a sample X1,..., Xn is i.i.d. against the alternative that the distributions of the Xi are stochastically increasing. Mann (1945) proposed the test which rejects for large values of N, where N is the number of pairs (Xi, X j) with i < j and Xi < X j . Determine the limiting distribution of N, suitably normalized. How large should we choose the critical value

(for large n) in order to control the Type 1 error at α?