638 Nonparametric and Robust StatisticsShow essentially thatrqc=
1
n{#(Xi,Yi)∈I+#(Xi,Yi)∈III−#(Xi,Yi)∈II−#(Xi,Yi)∈IV}.
(10.8.18)
Hence,rqcis referred to as thequadrant countcorrelation coefficient.
10.8.9.Set up the asymptotic test of independence usingrqcof the last exer-
cise. Then use it to test for independence between the 1500 m race times and the
marathon race times of the data in Example 10.8.1.10.8.10.Obtain the rank correlation coefficient when normal scores are used; that
is, the scores area(i)=Φ−^1 (i/(n+ 1)),i =1,...,n.CallitrN.Setupthe
asymptotic test of independence usingrN of the last exercise. Then use it to test
for independence between the 1500 m race times and the marathon race times of
the data in Example 10.8.1.
10.8.11.Suppose that the hypothesisH 0 concerns the independence of two random
variablesXandY. That is, we wish to testH 0 :F(x, y)=F 1 (x)F 2 (y), whereF, F 1 ,
andF 2 are the respective joint and marginal distribution functions of the continuous
type, against all alternatives. Let (X 1 ,Y 1 ),(X 2 ,Y 2 ),...,(Xn,Yn) be a random sam-
ple from the joint distribution. UnderH 0 , the order statistics ofX 1 ,X 2 ,...,Xnand
the order statistics ofY 1 ,Y 2 ,...,Ynare, respectively, complete sufficient statistics
forF 1 andF 2 .UserS,rqc,andrNto create an adaptive distribution-free test ofH 0.
Remark 10.8.2.It is interesting to note that in an adaptive procedure it would
be possible to use different score functions for theXsandYs. That is, the order
statistics of theX values might suggest one score function and those of theYs
another score function. Under the null hypothesis of independence, the resulting
procedure would produce anαlevel test.
10.9RobustConcepts
In this section, we introduce some of the concepts inrobustestimation. We intro-
duce these concepts for the location model discussed in Sections 10.1–10.3 of this
chapter and then apply them to the simple linear regression model of Section 10.7.
In a review article, McKean (2004) presents three introductory lectures on robust
concepts.10.9.1 LocationModel..........................
In a few words, we say an estimator isrobustif it is not sensitive to outliers in the
data. In this section, we make this more precise for the location model. Suppose
then thatX 1 ,X 2 ,...,Xnis a random sample which follows the location model as
given in Definition 10.1.2; i.e.,
Xi=θ+εi,i=1, 2 ,...,n, (10.9.1)