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Stat 411 – Homework 02
Due: Friday 01/27
1. Let
X
1
,...,X
n
be iid
Unif
(0
,θ
), where
θ >
0 is unknown.
(a) Let
ˆ
θ
n
=
X
(
n
)
, the sample maximum. Find the CDF of
ˆ
θ
n
. (Hint: This is
similar to Problem #2 on Homework 01.)
(b) Show that
ˆ
θ
n
is a consistent estimator of
θ
. (Hint:
θ
is always greater than
ˆ
θ
n
(why?); this makes it easy to use part (a) above.)
2. Same setup as in Problem #1 above.
(a) Show that
ˆ
θ
n
is not an unbiased estimator of
θ
. (Hint: Get the PDF of
ˆ
θ
n
by
diﬀerentiating the CDF in Problem #1a above, then calculate expected value
as usual. Alternatively, see the hint to Problem #4b below.)
(b) Use the work in part (a) to construct a new estimator
˜
θ
n
, a linear function
ˆ
θ
n
,
which is an unbiased estimator of
θ
. Show that
˜
θ
n
is also consistent.
3. Suppose
X
∼
Bin
(
n
= 10
,θ
) where
θ
∈
(0
,
1) is unknown. Consider the following
two estimators of
θ
:
ˆ
θ
1
=
X/
10 and
ˆ
θ
2
= (
X
+ 1)
/
12.
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This note was uploaded on 03/12/2012 for the course STAT 411 taught by Professor Staff during the Spring '08 term at Ill. Chicago.
 Spring '08
 STAFF

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