608 Nonparametric and Robust Statistics

We first define a general class of rank scores. Letφ(u) be a nondecreasing

function defined on the interval (0,1), such that

∫ 1

0 φ

(^2) (u)du <∞.Wecallφ(u)
a∫scorefunction. Without loss of generality, we standardize this function so that
1
0 φ(u)du=0and
∫ 1
0 φ
(^2) (u)du= 1; see Exercise 10.5.1. Next, define the scores
aφ(i)=φ[i/(n+1)], fori=1,...,n.Thenaφ(1)≤aφ(2)≤ ··· ≤aφ(n). As-
sume that
∑n
i=1a(i) = 0, (this essentially follows from
∫
φ(u)du= 0, see Exercise
10.5.12). Consider the test statistic
Wφ=
∑n^2
j=1
aφ(R(Yj)), (10.5.2)
whereR(Yj) denotes the rank ofYjin the combined sample ofnobservations. Since
the scores are nondecreasing, a natural rejection rule is given by
RejectH 0 in favor ofH 1 ifWφ≥c. (10.5.3)
Note that if we use the linear score functionφ(u)=
√
12(u−(1/2)), then
Wφ=
∑n^2
j=1
√
12
(
R(Yj)
n+1
−
1
2
)

√
12
n+1
∑n^2
j=1
(
R(Yj)−
n+1
2
)

√
12
n+1
W−
√
12 n 2
2
, (10.5.4)
whereW is the MWW test statistic, (10.4.5). Hence the special case of a linear
score function results in the MWW test statistic.
To complete the decision rule (10.5.2), we need the null distribution of the test
statisticWφ. But many of its properties follow along the same lines as that of the
MWW test. First,Wφis distribution free because, under the null hypothesis, every
subset of ranks for theYjs is equilikely. In general, the distribution ofWφcannot
be obtained in closed form, but it can be generated recursively similarly to the
distribution of the MWW test statistic. Next, to obtain the null mean ofWφ,use
the fact thatR(Yj) is uniform on the integers 1, 2 ,...,n. Because
∑n
i=1aφ(i)=0,
we then have
EH 0 (Wφ)=
∑n^2
j=1
EH 0 (aφ(R(Yj))) =
∑n^2
j=1
∑n
i=1
aφ(i)
1
n
=0. (10.5.5)
To determine the null variance, first define the quantitys^2 aby the equation
EH 0 (a^2 φ(R(Yj))) =
∑n
i=1
a^2 φ(i)
1
n

1
n
∑n
i=1
a^2 φ(i)=
1
n
s^2 a. (10.5.6)