Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

614 Nonparametric and Robust Statistics

section. First, though, note an invariance that simplifies matters. SupposeZis
a scale and location transformation of a random variableX; i.e.,Z=a(X−b),
wherea>0and−∞<b<∞. Because the efficacy varies indirectly with scale, we
havec^2 fZ=a−^2 c^2 fX. Furthermore, as Exercise 10.5.9 shows, the efficacy is invariant
to location and, also,I(fZ)=a−^2 I(fX). Hence the quantity maximized above is
invariant to changes in location and scale. In particular, in the derivation of optimal
scores, only the form of the density is important.
Example 10.5.1(Normal Scores). Suppose the error random variableεihas a
normal distribution. Based on the discussion in the last paragraph, we can take the
pdf of aN(0,1) distribution as the form of the density. So considerfZ(z)=φ(z)=
(2π)−^1 /^2 exp{− 2 −^1 z^2 }.Then−φ′(z)=zφ(z). Let Φ(z) denote the cdf ofZ. Hence
the optimal score function is

φN(u)=−κ

φ′(Φ−^1 (u))

φ(Φ−^1 (u))

=Φ−^1 (u); (10.5.29)

see Exercise 10.5.5, which shows thatκ= 1 as well as that

∫
φN(u)du=0. The
corresponding scores,aN(i)=Φ−^1 (i/(n+ 1)), are often called thenormal scores.
Denote the process by

WN(Δ) =

∑n^2

j=1

Φ−^1 [R(Yj−Δ)/(n+1)]. (10.5.30)

The associated test statistic for the hypotheses (10.5.1) is the statisticWN=WN(0).
The estimator of Δ solves the estimating equations

WN(Δ̂N)≈ 0. (10.5.31)

Although the estimate cannot be obtained in closed form, this equation is relatively
easy to solve numerically. From the above discussion, ARE(Δ̂N,Y−X)=1atthe
normal distribution. Hence normal score procedures are fully efficient at the normal
distribution. Actually, a much more powerful result can be obtained for symmet-
ric distributions. It can be shown that ARE(Δ̂N,Y−X)≥1 at all symmetric
distributions.

Example 10.5.2(Wilcoxon Scores).Suppose the random errors,εi,i=1, 2 ,...,n,
have a logistic distribution with pdffZ(z)=exp{−z}/(1 + exp{−z})^2. Then the
corresponding cdf isFZ(z)=(1+exp{−z})−^1. As Exercise 10.5.11 shows,

−f

Z′(z)

fZ(z)=FZ(z)(1−exp{−z})andF

− 1

Z (u)=log

u

1 −u. (10.5.32)

Upon standardization, this leads to the optimal score function,

φW(u)=

√

12(u−(1/2)), (10.5.33)

that is, the Wilcoxon scores. The properties of the inference based on Wilcoxon
scores are discussed in Section 10.4. LetΔ̂W =med{Yj−Xi}denote the corre-

sponding estimate. Recall that ARE(Δ̂W,Y−X)=0.955 at the normal. Hodges
and Lehmann (1956) showed that ARE(Δ̂W,Y−X)≥ 0 .864 over all symmetric
distributions.