Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

10.8. Measures of Association 637

a= 0. As in expression (10.5.6), lets^2 a=

∑n

i=1a

(^2) (i). Consider the rank correlation
coefficient,
ra=
1
s^2 a
∑n
i=1
a(R(Xi))a(R(Yi)). (10.8.16)
(a)Show thatrais a correlation coefficient on the items
{(a[R(X 1 )],a[R(Y 1 )]),(a[R(X 2 )],a[R(Y 2 )]),...,(a[R(Xn)],a[R(Yn)])}.
(b)For the score functionφ(u)=
√
12(u−(1/2)), show thatra=rS,Spearman’s
rho.
(c)Obtainrafor the sign score functionφ(u)=sgn(u−(1/2)). Call this rank
correlation coefficientrqc. (The subscriptqcis obvious from Exercise 10.8.8.)
10.8.6.Write an R function that computes a percentile bootstrap confidence inter-
val for Kendall’sτ. Run your function for the data discussed in Example 10.8.1 and
compare your answer with the confidence interval for Kendall’sτgiveninRemark
10.8.1.
Note: The following R code obtains resampled vectors ofxandy:
ind = 1:length(x); mat=cbind(x,y); inds=sample(ind,n,replace=T)
mats=mat[inds,]; xs=mats[,1]; ys=mats[,2]
10.8.7.Consider the general score rank correlation coefficientradefined in Exercise
10.8.5. Consider the null hypothesisH 0 :XandYare independent.
(a)Show thatEH 0 (ra)=0.
(b)Based on part (a) andH 0 , as a first step in obtaining the variance underH 0 ,
show that the following expression is true:
VarH 0 (ra)=
1
s^4 a
∑n
i=1
∑n
j=1
EH 0 [a(R(Xi))a(R(Xj))]EH 0 [a(R(Yi))a(R(Yj))].
(c)To determine the expectation in the last expression, consider the two cases
i=jandi =j. Then using uniformity of the distribution of the ranks, show
that
VarH 0 (ra)=
1
s^4 a
1
n− 1
s^4 a=
1
n− 1

. (10.8.17)

10.8.8.Consider the rank correlation coefficient given byrqcin part (c) of Exer-
cise 10.8.5. LetQ 2 XandQ 2 Y denote the medians of the samplesX 1 ,...,Xnand
Y 1 ,...,Yn, respectively. Now consider the four quadrants:

I = {(x, y):x>Q 2 X,y > Q 2 Y}

II = {(x, y):x<Q 2 X,y > Q 2 Y}

III = {(x, y):x<Q 2 X,y < Q 2 Y}

IV = {(x, y):x>Q 2 X,y < Q 2 Y}.