Some texts define the Wilcoxon signed rank statistic as S n = iRi, where i = 1 if Xi > 0 and 0 otherwise. Determine

Some texts define the Wilcoxon signed rank statistic as S˜

n = η˜iRi, where η˜i = 1 if Xi > θ0 and 0 otherwise. Determine the mean and variance of S˜

n, and show that tests based on Sn and on S˜

n are equivalent in the following sense: under a null hypothesis, Sn = S˜

n − S˜∗

n, where S˜

n and S˜∗ have the same distribution