574 Nonparametric and Robust StatisticsThe following symmetric decision rule seems appropriate:RejectH 0 in favor ofH 1 ifS(θ 0 )≤c 1 or ifS(θ 0 )≥n−c 1. (10.2.8)For a levelαtest,c 1 would be chosen such thatα/2=PH 0 (S(θ 0 )≤c 1 ). Recall
that thep-value is given byp̂=2min{PH 0 (S(θ 0 )≤s),PH 0 (S(θ 0 )≥s)},wheresis
the realized value ofS(θ 0 ) based on the sample.
Example 10.2.1(Shoshoni Rectangles).A golden rectangle is a rectangle in which
the ratio of the width (w) to the length (l) is the golden ratio, which is approxi-
mately 0.618. It can be characterized in various ways. For example,w/l=l/(w+l)
characterizes the golden rectangle. It is considered to be an aesthetic standard in
Western civilization and appears in art and architecture going back to the ancient
Greeks. It now appears in such items as credit and business cards. In a cultural
anthropology study, DuBois (1960) reports on a study of the Shoshoni beaded bas-
kets. These baskets contain beaded rectangles, and the question was whether the
Shoshonis use the same aesthetic standard as the West. LetXdenote the ratio of
the width to the length of a Shoshoni beaded basket. Letθbe the median ofX.
The hypotheses of interest are
H 0 : θ=0.618 versusH 1 : θ =0. 618.These are two-sided hypotheses. It followsfrom the above discussion that the sign
test rejectsH 0 in favor ofH 1 ifS(0.618)≤corS(0.618)≥n−c.
A sample of 20 width to length (ordered) ratios from Shoshoni baskets resulted
in the data
Width-to-Length Ratios of Rectangles
0.553 0.570 0.576 0.601 0.606 0.606 0.609 0.611 0.615 0.628
0.654 0.662 0.668 0.670 0.672 0.690 0.693 0.749 0.844 0.933The data can be found in the fileshoshoni.rda. For these data, the sign test statis-
tic isS(0.618) = 11. Using R thep-value is:2*(1-pbinom(10,20,.5))= 0.8238.
Thus there is no evidence to rejectH 0 based on these data.
A boxplot and a normalq−qplot of the data are given in Figure 10.2.1. Notice
that the data contain two, possibly three, potential outliers. The data do not appear
to be drawn from a normal distribution.
We next obtain several useful results concerning the power function of the sign
test for the hypotheses (10.2.2). The following function proves useful here and in
the associated estimation and confidence intervals described below. Define
S(θ)=#{Xi>θ}. (10.2.9)The sign test statistic is given byS(θ 0 ). We can easily describe the functionS(θ).
First, note that we can write it in terms of the order statisticsY 1 <···<Ynof
X 1 ,...,Xnbecause #{Yi>θ}=#{Xi>θ}.Nowifθ<Y 1 ,thenalltheYis
are larger thanθand, henceS(θ)=n.Next,ifY 1 ≤θ<Y 2 thenS(θ)=n−1.
Continuing this way, we see thatS(θ) is a decreasing step function ofθ,whichsteps