Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

616 Nonparametric and Robust Statistics

Table 10.5.2:Summary of analyses for Example 10.5.3

Method Test Statistic Standardized p-Value Estimate of Δ

Studentt Y−X=5.46 1.47 0.16 5.46

Wilcoxon W= 270 1.53 0.12 5.20

Normal scores WN=3. 73 1.48 0.14 5.15

Notice that the standardized tests statistics and their correspondingp-values are
quite similar and all would result in the same decision regarding the hypotheses.
As shown in the table, the corresponding point estimates of Δ are also alike.
We changedx 5 to be an outlier with value 95.5 and then reran the analyses. The
t-analysis was the most affected, for on the changed data,t=0.63 with ap-value
of 0.53. In contrast, the Wilcoxon analysis was the least affected (z=1.37 and
p=0.17). The normal scores analysis was more affected by the outlier than the
Wilcoxon analysis withz=1.14 andp=0.25.

Example 10.5.4(Sign Scores). For our final example, suppose that the ran-
dom errorsε 1 ,ε 2 ,...,εn have a Laplace distribution. Consider the convenient
formfZ(z)=2−^1 exp{−|z|}.ThenfZ′(z)=− 2 −^1 sgn(z)exp{−|z|}and, hence,
−fZ′(FZ−^1 (u))/fZ(FZ−^1 (u)) = sgn(z). ButFZ−^1 (u)>0 if and only ifu> 1 /2. The
optimal score function is
φS(u)=sgn

(

u−

1

2

)

, (10.5.34)

which is easily shown to be standardized. The corresponding process is

WS(Δ) =

∑n^2

j=1

sgn

[

R(Yj−Δ)−

n+1

2

]

. (10.5.35)

Because of the signs, this test statistic can be written in a simpler form, which is
often calledMood’stest; see Exercise 10.5.13.
We can also obtain the associated estimator in closed form. The estimator solves
the equation
∑n^2

j=1

sgn

[

R(Yj−Δ)−

n+1

2

]

=0. (10.5.36)

For this equation, we rank the variables

{X 1 ,...,Xn 1 ,Y 1 −Δ,...,Yn 2 −Δ}.

Because ranks, though, are invariant to a constant shift, we obtain the same ranks

if we rank the variables

X 1 −med{Xi},...,Xn 1 −med{Xi},Y 1 −Δ−med{Xi},...,Yn 2 −Δ−med{Xi}.

Therefore, the solution to equation (10.5.36) is easily seen to be

Δ̂S=med{Yj}−med{Xi}. (10.5.37)