Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

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