STA 2006
Confidence Intervals: Large Sample Approximation
20072008 Term II
1
Motivation
In the previous chapters,
X
1
, . . . , X
n
are i.i.d. sampled from
N
(
μ, σ
2
). The
C.I. of
μ
is derived by either considering
Z
=
¯
X

μ
σ/
√
n
∼
N
(0
,
1) (known
σ
case)
(1)
or
T
=
¯
X

μ
S/
√
n
∼
t
(
n

1) (Unknown
σ
case)
.
(2)
In this chapter, we relax the assumption of normal population. In general,
the distribution of
Z
can be complex for small sample. However, if
n
is large
and
X
1
, . . . , X
n
are i.i.d. (nonnormal) with common mean
μ
and variance
σ
2
, (1) is still true by the Central Limit Theorem (CLT).
Theorem 5.41 in HT:(CLT)
If
X
1
, . . . , X
n
are i.i.d. with common mean
μ
and variance
σ
2
, then
Z
n
=
√
n
(
¯
X
n

μ
)
σ
converges in distribution to
N
(0
,
1) as
n
→ ∞
. That is, for each
x
∈
R
,
lim
n
→∞
Pr
{
Z
n
≤
x
}
=
x
∞
1
√
2
π
exp
{
z
2
2
}
dz
= Φ(
x
)
,
(3)
or equivalently,
Z
n
d
→
Z
where
Z
∼
N
(0
,
1).
(or
Z
n
converges to
Z
in
distribution
.)
Remarks:
•
Application of CLT: if
n
is large,
Pr
{
Z
n
≤
z
} ≈
Φ(
z
).
•
The general definition of convergent in distribution:
Let
{
X
n
}
be a
sequence of random variables and
X
be a random variable. Also, sup
pose
F
n
and
F
are the c.d.fs of
X
n
and
X
. Now for any
x
in the set of
continuity points of
F
, if we have
lim
n
→∞
Pr
{
X
n
≤
x
}
= lim
n
→∞
F
n
(
x
) =
F
(
x
) = Pr
{
X
≤
x
}
,
then
X
n
is said to be converging in distribution to
X
.
Such fact is
denoted by
X
n
d
→
X
. To apply this fact, if
n
is large and
x
is in the
set of continuity points of
F
,
Pr
{
X
n
≤
x
}
=
F
n
(
x
)
≈
F
(
x
).
1
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Example 1:
Suppose
X
1
, . . . , X
n
are i.i.d. sampled from exponential distri
bution with mean 1
/λ
. Find the (approximate) 100(1

α
)% C.I. of
λ
when
n
is large.
Solution:
Note that
E
[
X
1
] = 1
/λ
and
V ar
(
X
1
) = 1
/λ
2
. By CLT,
√
n
(
¯
X
n

1
/λ
)
1
/λ
is approximately
N
(0
,
1) when
n
is large. That is, when
n
is large,
Pr
{
z
α/
2
≤
√
n
(
¯
X
n

1
/λ
)
1
/λ
≤
z
α/
2
} ≈
1

α
⇔
Pr
{

z
α/
2
√
n
+ 1
1
¯
X
n
≤
λ
≤
z
α/
2
√
n
+ 1
1
¯
X
n
} ≈
1

α
Therefore, the approximate 100(1

α
)% C.I. of
λ
is given by:
[

z
α/
2
√
n
+ 1
1
¯
x
n
,
z
α/
2
√
n
+ 1
1
¯
x
n
].
The approximation can be evaluated by using the simulation program in
”Exp.R” for
λ
= 1 and target confidence level 95% with 100,000 replicates.
n
5
10
20
50
100
500
Simulated C. L.
95
.
563%
95
.
482%
95
.
230%
95
.
176%
95
.
050%
95
.
017%
Example 2:
Suppose
I
1
, . . . , I
n
are i.i.d. sampled from Bernoulli distribu
tion with probability of success
p
. Find the (approximate) 100(1

α
)% C.I.
of
p
when
n
is large.
Solution:
Note that
E
[
I
1
] =
p
and
V ar
(
I
1
) =
p
(1

p
). By CLT,
√
n
(
¯
I
n

p
)
p
(1

p
)
is approximately
N
(0
,
1) when
n
is large. Note that
¯
I
n
is the same as ˆ
p
in
the overview chapter and it is the maximum likelihood estimate of
p
. When
n
is large, the C.I. of
p
is derived by using
Pr
{
z
α/
2
≤
√
n
(
¯
I
n

p
)
p
(1

p
)
≤
z
α/
2
} ≈
1

α
⇔
Pr
{
n
(
¯
I
n

p
)
2
≤
z
2
α/
2
p
(1

p
)
} ≈
1

α.
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 Spring '11
 Ho
 Statistics, Normal Distribution, Probability theory, Estimation theory, Xn, WLLN

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