522 Chapter 12:Nonparametric Hypothesis Tests
where
t∗=min(
t,n(n+1)
2−t)It remains to computePH 0 {T≤t∗}. To do so, letPk(i) denote the probability, under
H 0 , that the signed rank statisticTwill be less than or equal toiwhen the sample size isk.
We will determine a recursive formula forPk(i) starting withk=1. Whenk=1, since
there is only a single data value, which, whenH 0 is true, is equally likely to be either less
than or greater thanm 0 , it follows thatTis equally likely to be either 0 or 1. Thus
P 1 (i)=
0 i< 0
1
2 i=^0
1 i≥ 1(12.3.5)Now suppose the sample size isk. To computePk(i), we condition on the value ofIkas
follows:
Pk(i)=PH 0
∑kj= 1jIj≤i
=PH 0
∑kj= 1jIj≤i|Ik= 1
PH 0 {Ik= 1 }+PH 0
∑kj= 1jIj≤i|Ik= 0
PH 0 {Ik= 0 }=PH 0
k∑− 1j= 1jIj≤i−k|Ik= 1
PH^0 {Ik=^1 }+PH 0
k∑− 1j= 1jIj≤i|Ik= 0
PH 0 {Ik= 0 }=PH 0
k∑− 1j= 1jIj≤i−k
PH 0 {Ik= 1 }+PH 0
∑k−^1j= 1jIj≤i
PH 0 {Ik= 0 }where the last equality utilized the independence ofI 1 ,...,Ik− 1 , andIk(whenH 0 is
true). Now
∑k− 1
j= 1 jIjhas the same distribution as the signed rank statistic of a sample
of sizek−1, and since
PH 0 {Ik= 1 }=PH 0 {Ik= 0 }=^12