580 Nonparametric and Robust Statistics
and
σ^2 X(0) =V 0 (X)=
σ^2
n
. (10.2.22)
Thus, by (10.2.21) and (10.2.22), the efficacy of thet-test is
ct= lim
n→∞
μ′X(0)
√
n(σ/
√
n)
=
1
σ
. (10.2.23)
As confirmed in Exercise 10.2.9, the asymptotic power of the large sample levelα,
t-test under the sequence of alternatives (10.2.13) is 1−Φ(zα−δct). Thus we can
compare the sign andt-tests by comparing their efficacies. We do this from the
perspective of sample size determination.
Assume without loss of generality thatH 0 : θ= 0. Now suppose we want
to determine the sample size so that a levelαsign test can detect the alternative
θ∗>0 with (approximate) probabilityγ∗. That is, findnso that
γ∗=γ(θ∗)=Pθ∗
[
S(0)−(n/2)
√
n/ 2
≥zα
]
. (10.2.24)
Writeθ∗=
√
nθ∗/
√
n. Then, using the asymptotic power lemma, we have
γ∗=γ(
√
nθ∗/
√
n)≈ 1 −Φ(zα−
√
nθ∗τS−^1 ).
Now denotezγ∗to be the upper 1−γ∗quantile of the standard normal distribution.
Then, from this last equation, we have
zγ∗=zα−
√
nθ∗τS−^1.
Solving forn,weget
nS=
(
(zα−zγ∗)τS
θ∗
) 2
. (10.2.25)
As outlined in Exercise 10.2.9, for this situation the sample size determination for
the test based on the sample mean is
nX=
(
(zα−zγ∗)σ
θ∗
) 2
, (10.2.26)
whereσ^2 =Var(ε).
Suppose we have two tests of the same level for which the asymptotic power
lemma holds and for each we determine the sample size necessary to achieve power
γ∗at the alternativeθ∗. Then the ratio of these sample sizes is called theasymp-
totic relative efficiency(ARE) between the tests. We show later that this is the
same as the ARE defined in Chapter 6 between estimators. Hence the ARE of the
sign test to thet-test is
ARE(S, t)=
nX
nS
=
σ^2
τS^2
=
c^2 S
c^2 t
. (10.2.27)