Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

628 Nonparametric and Robust Statistics

Number of calls

0

5

10

15

20

50 55 60

Year

65 70

LS–Fit

Wilcoxon–Fit

Figure 10.7.1: Plot of telephone data, Example 10.7.2, overlaid with Wilcoxon

and LS fits.

see Exercise 10.7.2. Further, as Exercise 10.7.5 shows, this property implies that
the power curve for the one-sided tests ofH 0 : β= 0 are monotone, assuring the
unbiasedness of the tests based onTφ.
We can now derive the efficacy of the process. Let μT(β)=Eβ[T(0)] and
σT^2 (0) = Var 0 [T(0)]. Expression (10.7.5) gives the result forσ^2 T(0). Recall that for
the meanμT(β), we need its derivative at 0. We freely use the relationship between
rankings and the empirical cdf and then approximate this empirical cdf with the
true cdf. Hence

μT(β)=Eβ[T(0)] = E 0 [T(−β)] =

∑n

i=1

(xi−x)E 0 [aφ(R(Yi+xiβ))]

=

∑n

i=1

(xi−x)E 0

[

φ

(

nF̂n(Yi+xiβ)

n+1

)]

≈

∑n

i=1

(xi−x)E 0 [φ(F(Yi+xiβ))]

=

∑n

i=1

(xi−x)

∫∞

−∞

φ(F(y+xiβ))f(y)dy. (10.7.11)