592 Nonparametric and Robust StatisticsThis latter probability can be expressed as follows:P 0 (X 1 +X 2 >− 2 θn)=E 0 [P 0 (X 1 >− 2 θn−X 2 |X 2 )] =E 0 [1−F(− 2 θn−X 2 )]=∫∞−∞[1−F(− 2 θn−x)]f(x)dx=∫∞−∞F(2θn+x)f(x)dx≈∫∞−∞[F(x)+2θnf(x)]f(x)dx=
1
2+2θn∫∞−∞f^2 (x)dx, (10.3.23)wherewehaveusedthefactsthatX 1 andX 2 are iid and symmetrically distributed
about 0, and the mean value theorem. Hence
μT+(θn)≈an(
n
2)(
1
2+2θn∫∞−∞f^2 (x)dx). (10.3.24)
Putting (10.3.18) and (10.3.24) together, we have the efficacy
cT+= lim
n→∞μ′T+(0)
√
nσT+(0)=√
12∫∞−∞f^2 (x)dx. (10.3.25)In a more advanced text, this development can be made into a rigorous argument
for the following asymptotic power lemma.
Theorem 10.3.2(Asymptotic Power Lemma).Consider the sequence of hypotheses
(10.3.16). The limit of the power function of the large sample, sizeα, signed-rank
Wilcoxon test is given by
lim
n→∞
γSR(θn)=1−Φ(zα−δτW−^1 ), (10.3.26)whereτW =1/[
√
12∫∞
−∞f(^2) (x)dx]is the reciprocal of the efficacyc
T+andΦ(z)is
the cdf of a standard normal random variable.
As shown in Exercise 10.3.10, the parameterτWis a scale functional.
The arguments used in the determination of the sample size in Section 10.2 for
the sign test were based on the asymptotic power lemma; hence, these arguments
follow almost verbatim for the signed-rank Wilcoxon. In particular, the sample size
needed so that a levelαsigned-rank Wilcoxon test of the hypotheses (10.3.1) can
detect the alternativeθ=θ 0 +θ∗with approximate probabilityγ∗is
nW=
(
(zα−zγ∗)τW
θ∗
) 2
. (10.3.27)
Using (10.2.26), the ARE between the signed-rank Wilcoxon test and thet-test
basedonthesamplemeanis
ARE(T,t)=
nt
nT=
σ^2
τW^2. (10.3.28)