S Wilcoxon’s signed-rank test. Let X1; : : : ; Xn be independent real-valued random variables with identical distribution Q on .R;B/. Assume Q is continuous and symmetric with respect to 0, i.e., FQ.c/ D 1 FQ.c/ for all c 2 R. For each 1 i n, let Zi D 1¹Xi>0º

and let R C

i be the rank of jXi j in the sequence jX1j; : : : ; jXnj of absolute observations. Let W C D

Pn iD1 Zi R C

i be the corresponding signed-rank sum. Show the following:

(a) For each i , Zi and jXi j are independent.

(b) The random vector RC D .R C

1 ; : : : ; RC n / is independent of the random set Z D ¹1 i n W Zi D 1º :

(c) Z is uniformly distributed on the power set Pn of ¹1; : : : ; nº, and RC is uniformly distributed on the permutation set Sn.

(d) For each 0 l n.nC1/=2, it is the case that P.WC D l/ D 2n N.lI n/ for

(The N.lI n/ can be determined by combinatorial means, and an analogue of the central limit theorem (11.28) holds. Therefore, W C can be used as a test statistic for the null hypothesis H0 W Q is symmetric, which is suitable for the experimental design of matched pairs.)

= N(l; n) {AC{1,....n}: i=1}. iA