NTHU MATH 2820, 2008, Lecture Notes
Ch8, p.9
•
Questions:
1. Is the estimate (i.e., 24.9) the real
λ
?
2. Next time when the same procedure (same sample size,
same estimator, same
. . .
) is repeated again to get a
new estimate, how far the future estimate will be away
from 24.9?
3. In which range will you expect say 95% of the future
estimate falls? (e.g., [24.8, 25] or [15, 35]?)
4. This is a question related to the stability/uncertainty/
variation of the estimator.
5. How to characterize the stability of an estimator? (
Note.
We have to answer the question .)
To evaluate the stability/uncertainty of an
estimation procedure, it is required to
know what the sampling distribution is.
made by Shao-Wei Cheng (NTHU, Taiwan)
Ch8, p.10
Example 6.5
(cont. Ex.6.4, TBp. 262)
•
Exact sampling distribution
of
ˆ
λ
:
Because
X
1
, X
2
, . . . , X
n
i.i.d.
∼
P
(
λ
),
S
=
n
i
=1
X
i
∼
P
(
n
λ
)
E(
ˆ
λ
) =
1
n
E
(
S
) =
λ
,
Var(
ˆ
λ
) =
1
n
2
Var(
S
) =
λ
n
.
Thus, (1) the sampling distribution of
ˆ
λ
is
1
n
P
(
n
λ
), (2)
ˆ
λ
is unbiased,
and (3) the standard error of
ˆ
λ
is
σ
ˆ
λ
=
λ
/n.
•
Estimated standard error of
ˆ
λ
is
s
ˆ
λ
=
ˆ
λ
/n
=
24
.
9
/
23 = 1
.
04
.
•
Asymptotical method
: By CLT, sampling distribution of
ˆ
λ
is ap-
proximately normal when
n
is large enough.
Since a normally dis-
tributed random variable is very unlikely to be more than 2 standard
deviation away from its mean, errors in
ˆ
λ
is very unlikely to be more
than 2.08.