10.2. Sample Median and the Sign Test 575
0.9
0.8
0.7
0.6
Width to length ratios
Panel A
0.9
0.8
0.7
0.6
- 1.5
Width to length ratios
Panel B
- 0.5 0.5
Normal quantiles
1.0 1.5
Figure 10.2.1:Boxplot (Panel A) and normalq−qplot (Panel B) of the Shoshoni
data.
down one unit at each order statisticYi, attaining its maximum and minimum values
nand 0 atY 1 andYn, respectively. Figure 10.2.2 depicts this function.
We need the following translation property. Because we can always subtractθ 0
from eachXi, we can assume without loss of generality thatθ 0 =0.
Lemma 10.2.1.For everyk,
Pθ[S(0)≥k]=P 0 [S(−θ)≥k]. (10.2.10)
Proof:Note that the left side of equation (10.2.10) concerns the probability of the
event #{Xi> 0 },whereXihas medianθ. The right side concerns the probability
of the event #{(Xi+θ)> 0 }, where the random variableXi+θhas medianθ
(because underθ=0,Xihas median 0). Hence the left and right sides give the
same probability.