Consider again the conditions of Exercise 1. This time, let Di = Xi − Yi .Wilcoxon 1945) developed the following test of the hypotheses (10.8.7). Order the absolute values |D1|, . . . , |Dn| from smallest to largest, and assign ranks from 1 to n to the values. Then SW is set equal to the sum of all the ranks of those |Di| such that Di > 0. If p = Pr(Xi ≤ Yi) = 1/2, then he mean and variance of SW are
E(SW) = n(n + 1)/4, (10.8.8)
Var(SW) = n(n + 1)(2n + 1)/24. (10.8.9)
The test rejects H0 if SW ≥ c, where c is chosen to make the test have level of significance α0. This test is called the Wilcoxon signed ranks test. If n is large, a normal Distribution approximation allows us to use c = E(SW) + Φ−1(1− α0) Var(SW)1/2.
a. Let Wi = 1 if Xi ≤ Yi , and Wi = 0 if not. Show that
b. Prove that E(SW) is as stated in Eq. (10.8.8) under the assumption that p = 1/2. Hint: You may wish to use Eq. (4.7.13).
c. Prove that Var(SW) is as stated in Eq. (10.8.9) under the assumption that p = 1/2. Hint: You may wish to use Eq. (4.7.14).