Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

626 Nonparametric and Robust Statistics

i=1,...,n 1 ,andZn 1 +i=Yi,fori=n 1 +1,...,n,wheren=n 1 +n 2 .Letcibe

0 or 1 depending on whether 1≤i≤n 1 orn 1 +1≤i≤n. Then we can write the

two sample location models as

Zi=α+Δci+εi, (10.7.3)

whereε 1 ,ε 2 ,...,εnare iid with cdfF(x). Hence the shift in locations is the slope
parameter from this viewpoint.

Suppose the regression model (10.7.1) holds and, further, thatH 0 is true. Then
we would expect thatYiandxi−xare not related and, in particular, that they
are uncorrelated. Hence one could consider

∑n
i=1(xi−x)Yias a test statistic. As
Exercise 9.6.11 of Chapter 9 shows, if we additionally assume that the random
errorsεiare normally distributed, this test statistic, properly standardized, is the
likelihood ratio test statistic. Reasoning in the same way, for a specified score
function we would expect thataφ(R(Yi)) andxi−xare uncorrelated, underH 0.
Therefore, consider the test statistic

Tφ=

∑n

i=1

(xi−x)aφ(R(Yi)), (10.7.4)

whereR(Yi) denotes the rank ofYiamongY 1 ,...,Ynandaφ(i)=φ(i/(n+1)) for a
nondecreasing score functionφ(u) that is standardized, so that

∫
∫ φ(u)du=0and
φ^2 (u)du=1. ValuesofTφclose to 0 indicateH 0 is true.
AssumeH 0 is true. ThenY 1 ,...,Ynare iid random variables. Hence any per-
mutation of the integers{ 1 , 2 ,...,n}is equilikely to be the ranks ofY 1 ,...,Yn.So
the distribution ofTφis free ofF(x). Note that the distribution does depend on
x 1 ,x 2 ,...,xn. Thus, tables of the distribution are not available, although with high-
speed computing, this distribution can be generated. BecauseR(Yi) is uniformly
distributed on the integers{ 1 , 2 ,...,n}, it is easy to show that the null expectation
ofTφis zero. The null variance follows that ofWφof Section 10.5, so we have left
the details for Exercise 10.7.4. To summarize, the null moments are given by

EH 0 (Tφ)=0 and VarH 0 (Tφ)=n^1 − 1 s^2 a

∑n

i=1

(xi−x)^2 , (10.7.5)

wheres^2 ais the mean sum of the squares of the scores (10.5.6). Also, it can be shown
that the test statistic is asymptotically normal. Therefore, an asymptotic levelα
decision rule for the hypotheses (10.7.2) with the two-sided alternative is given by

RejectH 0 in favor ofH 1 if|z|=

∣

∣

∣

∣

∣

q Tφ

VarH 0 (Tφ)

∣

∣

∣

∣

∣

≥zα/ 2. (10.7.6)

The associated process is given by

Tφ(β)=

∑n

i=1

(xi−x)aφ(R(Yi−xiβ)). (10.7.7)