θ
0
)
τ
V
−
1
n
(
ˆ
θ
n
−
θ
0
)
>χ
2
k,α
parenrightBig
→
1
.
Example 6.27.
Let
X
1
,...,X
n
be i.i.d. random variables from a symmetric
c.d.f.
F
having finite variance and positive
F
′
.
Consider the problem of
testing
H
0
:
F
is symmetric about 0 versus
H
1
:
F
is not symmetric about
0. Under
H
0
, there are many estimators satisfying (6.92). We consider the
following five estimators:
(1)
ˆ
θ
n
=
¯
X
and
θ
=
E
(
X
1
);
(2)
ˆ
θ
n
=
ˆ
θ
0
.
5
(the sample median) and
θ
=
F
−
1
(
1
2
) (the median of
F
);
(3)
ˆ
θ
n
=
¯
X
a
(the
a
-trimmed sample mean defined by (5.77)) and
θ
=
T
(
F
),
where
T
is given by (5.46) with
J
(
t
) = (1
−
2
a
)
−
1
I
(
a,
1
−
a
)
(
t
),
a
∈
(0
,
1
2
);
(4)
ˆ
θ
n
= the Hodges-Lehmann estimator (Example 5.8) and
θ
=
F
−
1
(
1
2
);
(5)
ˆ
θ
n
=
W/n
−
1
2
, where
W
is given by (6.83) with
J
(
t
) =
t
, and
θ
=
T
(
F
)
−
1
2
with
T
given by (5.53).
Although the
θ
’s in (1)-(5) are different in general, in all cases
θ
= 0 is
equivalent to that
H
0
holds.
For
¯
X
, it follows from the CLT that (6.92) holds with
V
n
=
σ
2
/n
for any
F
, where
σ
2
= Var(
X
1
). From the SLLN,
S
2
/n
is a consistent estimator of
V
n
for any
F
. Thus, the test having rejection region (6.93) with
ˆ
θ
n
=
¯
X
and
V
n
replaced by
S
2
/n
is asymptotically correct. This test is asymptotically
equivalent to the one-sample t-test derived in
§
6.2.3.
From Theorem 5.10,
ˆ
θ
0
.
5
satisfies (6.92) with
V
n
= 4
−
1
[
F
′
(
θ
)]
−
2
n
−
1
for
any
F
. A consistent estimator of
V
n
can be obtained using the bootstrap
method considered in
§
5.5.3.
Another consistent estimator of
V
n
can be
obtained using Woodruff’s interval introduced in
§
7.4 (see Exercise 86 in
§
7.6). The test having rejection region (6.93) with
ˆ
θ
n
=
ˆ
θ
0
.
5
and
V
n
replaced
by a consistent estimator is asymptotically correct.
It follows from the discussion in
§
5.3.2 that
¯
X
a
satisfies (6.92) for any
F
.
A consistent estimator of
V
n
can be obtained using formula (5.110)
or the jackknife method in
§
5.5.2. The test having rejection region (6.93)
with
ˆ
θ
n
=
¯
X
a
and
V
n
replaced by a consistent estimator is asymptotically
correct.
From Example 5.8, the Hodges-Lehmann estimator satisfies (6.92) for
any
F
and
V
n
= 12
−
1
γ
−
2
n
−
1
under
H
0
, where
γ
=
integraltext
F
′
(
x
)
dF
(
x
).
A

454
6. Hypothesis Tests
consistent estimator of
V
n
under
H
0
can be obtained using the result in
Exercise 102 in
§
5.6. The test having rejection region (6.93) with
ˆ
θ
n
= the
Hodges-Lehmann estimator and
V
n
replaced by a consistent estimator is
asymptotically correct.
Note that all tests discussed so far are not of limiting size
α
, since the
distributions of
ˆ
θ
n
are still unknown under
H
0
.
The test having rejection region (6.93) with
ˆ
θ
n
=
W/n
−
1
2
and
V
n
=
(12
n
)
−
1
is equivalent to the one-sample Wilcoxon signed rank test and is
shown to have limiting size
α
(
§
6.5.1). Also, (6.92) is satisfied for any
F
(
§
5.2.2).
Although Theorem 6.12 is not applicable, a modified proof of
Theorem 6.12 can be used to show the consistency of this test (exercise).