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Confidence Intervals
•
find functions
of
data
A
(
X
1
, . . . , X
2
) and
B
(
X
1
, . . . , X
n
) such that
P
θ
0
(
A
(
X
1
, . . . , X
n
)
≤
θ
0
≤
B
(
X
1
, . . . , X
n
)) = 1
−
α
•
then [
A
(
X
1
, . . . , X
n
)
, B
(
X
1
, . . . , X
n
)] is
a 1
−
α
confidence interval for
θ
•
for given significance level 1
−
α
many possible valid confidence intervals.
In order to construct confidence intervals, usually proceed as
follows:
1. find
a
(
θ
0
) and
b
(
θ
0
) such
that
P
θ
0
(
a
(
θ
0
)
≤
T
(
X
1
, . . . , X
n
)
≤
b
(
θ
0
)) = 1
−
α
for some
statistic
T
(
X
1
, . . . , X
n
) (typically will
use an estimator
θ
ˆ
here).
2.
rewrite
the event inside the probability in form
P
(
A
(
X
1
, . . . , X
n
)
≤
θ
0
≤
B
(
X
1
, . . . , X
n
)) = 1
−
α
3. evaluate
A
( ) and
B
( ) at the sample values
X
1
, . . . , X
n
to
obtain confidence interval
·
·
Some
important cases:
θ
ˆ
unbiased
and
normally distributed, Var(
θ
ˆ
) known:
•
[
A
(
X
1
, . . . , X
n
)
, B
(
X
1
, . . . , X
n
)] =
θ
ˆ
+ Φ
−
1
�
α
�
�
α
�
Var(
θ
ˆ
)
, θ
ˆ
+ Φ
−
1
1
−
Var(
θ
ˆ
)
2
2
θ
ˆ
unbiased and
normally distributed, Var(
θ
ˆ
) not known,
but have estimator
S
ˆ
:
•
[
A
(
X
1
, . . . , X
n
)
, B
(
X
1
, . . . , X
n
)] =
θ
ˆ
+
t
n
−
1
�
α
�
�
α
�
Var(
θ
ˆ
)
, θ
ˆ
+
t
n
−
1
1
−
Var(
θ
ˆ
)
2
2
θ
ˆ
not
normal,
n >
30
or
so: most estimators
we have seen so
far
turn out to
be asymptotically
•
normally distributed, so