624 Nonparametric and Robust Statistics
X 1 ,...,Xn 1 ,Y 1 ,...,Yn 2 are not identically distributed. There are adaptive proce-
dures based on residualsX 1 ,...,Xn 1 ,Y 1 −Δ̂,...,Yn 2 −Δ, wherê Δisaninitial̂
estimator of Δ; see page 237 of Hettmansperger and McKean (2011) for discussion
and Section 7.6 of Kloke and McKean (2014) for an R implementation.
EXERCISES
10.6.1.In Exercises 10.6.2 and 10.6.3, the student is asked to apply the adaptive
procedure described in Example 10.6.1 to real data sets. The hypotheses of interest
are
H 0 :Δ=0versusH 1 :Δ> 0 ,
where Δ =μY−μX. The four distribution-free test statistics are
Wi=
∑n^2
j=1
ai[R(Yj)],i=1, 2 , 3 , 4 , (10.6.4)
where
ai(j)=φi[j/(n+1)],
and the score functions are given by
φ 1 (u)= 2 u− 1 , 0 <u< 1
φ 2 (u)=sgn(2u−1), 0 <u< 1
φ 3 (u)=
⎧
⎨
⎩
4 u− 10 <u≤^14
0 14 <u≤^34
4 u− 3 34 <u< 1
φ 4 (u)=
{
4 u−(3/2) 0<u≤^12
1 / 2 12 <u< 1.
Note that we have adjusted the fourth scoreφ 4 (u) in Figure 10.6.1 so that it inte-
grates to 0 over the interval (0,1).
The theory of Section 10.5 states that, underH 0 , the distribution ofWiis
asymptotically normal with mean 0 and variance
VarH 0 (Wi)=
n 1 n 2
n− 1
⎡
⎣^1
n
∑n
j=1
a^2 i(j)
⎤
⎦.
Note, however, that the scores have not been standardized, so their squares integrate
to 1 over the interval (0,1). Hence, do not replace the term in brackets by 1. If
n 1 =n 2 = 15, find VarH 0 (Wi), fori=1,...,4.
10.6.2.Consider the data in Example 10.5.3 and the hypotheses
H 0 :Δ=0versusH 1 :Δ> 0 ,
where Δ =μY−μX. Apply the adaptive procedure described in Example 10.6.1
with the tests defined in Exercise 10.6.1 to test these hypotheses. Obtain thep-value
of the test.