Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

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.