5.9 Let D1,D2, . . . ,DN be a random sample of N nonzero observations from some continuous population which is symmetric with median zero.

Define jDij ¼

Xi if Di > 0 Yi if Di < 0



Assume there are m X values and n Y values, where mþn¼N and the X and Y values are independent. Show that the signed-rank test statistic Tþ calculated for these Di is equal to the sum of the ranks of the X observations in the combined ordered sample of m X’s and n Y’s and also that TþT is the sum of the X ranks minus the sum of the Y ranks. This sum of the ranks of the X’s is the test criterion for the Wilcoxon rank sum statistic in the two-sample problem to be discussed in Chapter 8. Show how Tþ might be used to test the hypothesis that the