590 Nonparametric and Robust StatisticsTable 10.3.1:Signed Ranks for Darwin Data, Example 10.3.1Cross- Self- Signed-
Pot Fertilized Fertilized Difference Rank
1 23.500 17.375 6.125 11
2 12.000 20.375 − 8. 375 − 14
3 21.000 20.000 1.000 2
4 22.000 20.000 2.000 4
5 19.125 18.375 0.750 1
6 21.550 18.625 2.925 5
7 22.125 18.625 3.500 7
8 20.375 15.250 5.125 9
9 18.250 16.500 1.750 3
10 21.625 18.000 3.625 8
11 23.250 16.250 7.000 12
12 21.000 18.000 3.000 6
13 22.125 12.750 9.375 15
14 23.000 15.500 7.500 13
15 12.000 18.000 − 6. 000 − 10value ofT+along with thep-value. The computed values are the same as those
computed above.There is another formulation ofT+which is useful for obtaining the properties
of the Wilcoxon signed-rank test and confidence intervals forθ.LetXi>0and
consider allXj such that−Xi<Xj <Xi. Thus all the averages (Xi+Xj)/2,
under these restrictions, are positive, including (Xi+Xi)/2. From the restriction,
though, the number of these positive averages is simply theR|Xi|.Doingthisfor
allXi>0, we obtain
T+=#i≤j{(Xj+Xi)/ 2 > 0 }. (10.3.14)
The pairwise averages (Xj+Xi)/2areoftencalledtheWalsh averages. Hence the
signed-rank Wilcoxon can be obtained by counting the number of positive Walsh
averages.
Based on the identity (10.3.14), we obtain the corresponding process. Let
T+(θ)=#i≤j{[(Xj−θ)+(Xi−θ)]/ 2 > 0 }=#i≤j{(Xj+Xi)/ 2 >θ}.(10.3.15)The process associated withT+(θ) is much like the sign process, (10.2.9). Let
W 1 <W 2 <···<Wn(n+1)/ 2 denote then(n+1)/2 ordered Walsh averages. Then a
graph ofT+(θ) would appear as in Figure 10.2.2, except the ordered Walsh averages
would be on the horizontal axis and the largest value on the vertical would be
n(n+1)/2. Hence the functionT+(θ) is a decreasing step function ofθ,which
steps down one unit at each Walsh average. This observation greatly simplifies the
discussion on the properties of the signed-rank Wilcoxon.