Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

622 Nonparametric and Robust Statistics

are

Wi=

∑n^2

j=1

ai[R(Yj)],i=1, 2 , 3 , 4 , (10.6.1)

where

ai(j)=φi[j/(n+1)],

and the four functions are displayed in Figure 10.6.1. The score functionφ 1 (u)
is the Wilcoxon. The score functionφ 2 (u) is the sign score function. The score
functionφ 3 (u) is good for short-tailed distributions, andφ 4 (u) is good for long,
right-skewed distributions with shift alternatives.

u

1

1 (u)

u

1

2 (u)

u

1

3 (u)

u

1

4 (u)

Figure 10.6.1:Plots of the score functionsφ 1 (u),φ 2 (u),φ 3 (u), andφ 4 (u).

We combine the two samples into one denoting the order statistics of the com-
bined sample byV 1 <V 2 <···<Vn. These are complete sufficient statistics for
F(x) under the null hypothesis. Fori=1,...,4, the test statisticWiis distribution
free underH 0 and, in particular, the distribution ofWidoes not depend onF(x).
Therefore, eachWiis independent ofV 1 ,V 2 ,...,Vn. We use a pair of selector statis-
tics (Q 1 ,Q 2 ), which are functions ofV 1 ,V 2 ,...,Vn, and hence are also independent
of eachWi. The first is

Q 1 =

U. 05 −M. 5

M. 5 −L. 05

, (10.6.2)

whereU. 05 ,M. 5 ,andL. 05 are the averages of the largest 5% of theVs, the middle
50% of theVs, and the smallest 5% of theVs, respectively. IfQ 1 is large (say 2
or more), then the right tail of the distribution seems longer than the left tail; that
is, there is an indication that the distribution is skewed to the right. On the other
hand, ifQ 1 <^12 , the sample indicates that the distribution may be skewed to the