580 Nonparametric and Robust Statisticsand
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 haveγ∗=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 isnX=(
(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)