Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

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.