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STAT 410
Homework #8
Fall 2011
(due Friday, October 28, by 3:00 p.m.)
1.
4.3.16
Hint:
+
+
+
=
1
2
1
2
n
n
t
n
t
e
o
n
t
for large
n
.
2.
4.3.17
Hint:
Use
Theorem 4.3.9.
3.
Let
Y
n
be
χ
2
(
n
).
What is the limiting distribution of
Z
n
=
n
n
Y

?
Hint:
We already know that
( )
n
n
n
2
Y

=

1
Y
2
n
n
n
( )
1
,
0
N
D
→
.
4.
Let
X
1
, X
2
, … , X
n
be a random sample from the distribution with probability
density function
( ) ( ) ( )
θ
1
1
θ
θ
;
X
x
x
f

+
=
⋅
,
0 <
x
< 1,
θ
> – 1.
Recall:
θ
~
=
2
X
1

is the method of moments estimator of
θ
.
Show that
θ
~
is asymptotically normally distributed
(
as
n
→
∞
).
Find the parameters.
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View Full Document5.
Let
X
1
, X
2
, … , X
n
be a random sample from the distribution with probability
density function
( ) ( ) ( )
θ
1
1
θ
θ
;
X
x
x
f

+
=
⋅
,
0 <
x
< 1,
θ
> – 1.
a)
What is the probability distribution of
Y =
( )
∑
=


n
i
i
1
X
1
ln
?
b)
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 Fall '08
 Monrad
 Statistics, Probability

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