Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

10.4. Mann–Whitney–Wilcoxon Procedure 599

In the shift case, the parameter Δ is independent of what location functional
is used. To see this, suppose we select an arbitrary location functional forX,say,
T(FX). Then we can writeXias

Xi=T(FX)+εi, (10.4.2)

whereε 1 ,...,εn 1 are iid withT(Fε) = 0. By (10.4.1) it follows that

Yj=T(FX)+Δ+εj,j=1, 2 ,...,n 2. (10.4.3)

HenceT(FY)=T(FX) + Δ. Therefore, Δ =T(FY)−T(FX) for any location
functional; i.e., Δ is the same no matter what functional is chosen to model location.
Assume then that the shift model, (10.4.1), holds for the two samples. Alterna-
tives of interest are the usual one- and two-sided alternatives. For convenience we
pick on the one-sided hypotheses given by

H 0 :Δ=0versusH 1 :Δ> 0. (10.4.4)

The exercises consider the other hypotheses. UnderH 0 , the distributions ofX
andY are the same, and we can combine the samples to have one large sample of
n=n 1 +n 2 observations. Suppose we rank the combined samples from 1 tonand
consider the statistic

W=

∑n^2

j=1

R(Yj), (10.4.5)

whereR(Yj) denotes the rank ofYj in the combined sample ofnitems. This
statistic is often called theMann–Whitney–Wilcoxon(MWW) statistic. Under
H 0 the ranks are uniformly distributed between theXisandtheYjs; however, under
H 1 :Δ>0, theYjs should get most of the large ranks. Hence an intuitive rejection
rule is given by
RejectH 0 in favor ofH 1 ifW≥c. (10.4.6)
We now discuss the null distribution ofW, which enables us to selectcfor
the decision rule based on a specified levelα. UnderH 0 , the ranks of theYjsare

(equilikely to be any subset of sizen^2 from a set ofnelements. Recall that there are
n
n 2

)
such subsets; therefore, if{r 1 ,...,rn 2 }is a subset of sizen 2 from{ 1 ,...,n},
then

P[R(Y 1 )=r 1 ,...,R(Yn 2 )=rn 2 ]=

(

n

n 2

)− 1

. (10.4.7)

This implies that the statisticWis distribution free underH 0. Although the null
distribution ofWcannot be obtained in closed form, there are recursive algorithms
which obtain this distribution; see Chapter 2 of the text by Hettmansperger and
McKean (2011). In the same way, the distribution of a single rankR(Yj) is uniformly
distributed on the integers{ 1 ,...,n}, underH 0. Hence we immediately have

EH 0 (W)=

∑n^2

j=1

EH 0 (R(Yj)) =

∑n^2

j=1

∑n

i=1

i

1

n

=

∑n^2

j=1

n(n+1)

2 n

=

n 2 (n+1)

2

.