STAT 409
Homework #6
Fall 2009
(due Friday, October 9, by 4:00 p.m.)
1.
Let
a
> 0,
θ
> 0
and
let
X
1
, X
2
, … , X
n
be a random sample of size
n
from a
distribution with probability density function
( 29 ( 29
1
X
X
θ
θ
;

⋅
=
=
a
a
x
a
x
f
x
f
,
0 <
x
<
θ
.
Suppose
a
is known.
a)
Find the sufficient statistic
Y =
u
(
X
1
, X
2
, … , X
n
)
for
θ
.
b)
Find the method of moments estimator
θ
~
of
θ
.
2.
Let
a
> 0,
θ
> 0
and
let
X
1
, X
2
, … , X
n
be a random sample of size
n
from a
distribution with probability density function
( 29 ( 29
1
X
X
θ
θ
;

⋅
=
=
a
a
x
a
x
f
x
f
,
0 <
x
<
θ
.
Suppose
a
is known.
a)
Find the maximum likelihood estimator
θ
ˆ
of
θ
.
b)
Is
θ
ˆ
a consistent estimator of
θ
?
Justify your answer
.
Hint:
Let
ε
> 0.
Find
P
(

θ
ˆ
–
θ

≥
ε
)
=
P
(
θ
ˆ
≤
θ
–
ε
)
+
P
(
θ
ˆ
≥
θ
+
ε
).
3.
Let
a
> 0,
θ
> 0
and
let
X
1
, X
2
, … , X
n
be a random sample of size
n
from a
distribution with probability density function
( 29 ( 29
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 Spring '11
 Stepanov
 Normal Distribution, Probability, probability density function, θ

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