578 Nonparametric and Robust Statistics
θn, we can approximate the mean of this test as follows:
Eθn
[
1
√
n
(
S(0)−
n
2
)]
= E 0
[
1
√
n
(
S(−θn)−
n
2
)]
=
1
√
n
∑n
i=1
E 0 [I(Xi>−θn)]−
√
n
2
=
1
√
n
∑n
i=1
P 0 (Xi>−θn)−
√
n
2
=
√
n
(
1 −F(−θn)−
1
2
)
=
√
n
(
1
2
−F(−θn)
)
≈
√
nθnf(0) =δf(0), (10.2.14)
where the step to the last line is due to the mean value theorem. It can be shown
in more advanced texts that the variance of [S(0)−(n/2)]/(
√
n/2) converges to 1
underθn, just as underH 0 ,andthat,furthermore,[S(0)−(n/2)−
√
nδf(0)]/(
√
n/2)
has a limiting standard normal distribution. This leads to theasymptotic power
lemma, which we state in the form of a theorem.
Theorem 10.2.2(Asymptotic Power Lemma).Consider the sequence of hypotheses
(10.2.13). The limit of the power function of the large sample, sizeα,signtestis
lim
n→∞
γ(θn)=1−Φ(zα−δτS−^1 ), (10.2.15)
whereτS=1/[2f(0)]andΦ(z)is the cdf of a standard normal random variable.
Proof: Using expression (10.2.14) and the discussion that followed its derivation,
we have
γ(θn)=Pθn
[
n−^1 /^2 [S(0)−(n/2)]
1 / 2
≥zα
]
= Pθn
[
n−^1 /^2 [S(0)−(n/2)−
√
nδf(0)]
1 / 2
≥zα−δ 2 f(0)
]
→ 1 −Φ(zα−δ 2 f(0)),
which was to be shown.
As shown in Exercise 10.2.5, the parameterτS=1/[2f(0)] is a scale parameter
(functional) as defined in Exercise 10.1.4 of the last section. We later show that
τS/
√
nis the asymptotic standard deviation of the sample median.
Note that there were several approximations used in the proof of Theorem 10.2.2.
A rigorous proof can be found in more advanced texts, such as those cited in Section
10.1. It is quite helpful for the next sections to reconsider the approximation of the