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Spring 2008
Homework #8
(due Friday, March 14, by 3:00 p.m.)
1.
Let X
1
, X
2
, … , X
n
be a random sample of size
n
from the distribution with
probability density function
( ) ( ) ( )
2
X
X
ln
1
;
x
x
x
f
x
f
⋅

=
=
,
x
> 1,
θ
> 1.
a)
Find the maximum likelihood estimator
ˆ
of
θ
.
b)
Suppose
θ
> 2. Find the method of moments estimator
~
of
θ
.
2.
If the random variable Y denotes an individual’s income, Pareto’s law claims that
P
(
Y
≥
y
) =
±
²
³
´
µ
y
k
, where
k
is the entire population’s minimum income. It
follows that
f
Y
(
y
) =
1
1
+
±
²
³
³
´
µ
y
k
,
y
≥
k
;
θ
≥
1.
The income information has been collected on a random sample of
n
individuals:
Y
1
, Y
2
, … , Y
n
. Assume
k
is known.
a)
Find the maximum likelihood estimator
ˆ
of
θ
.
b)
Find the method of moments estimator
~
of
θ
.
3.
Let Y
1
, Y
2
, … , Y
n
be a random sample of size
n
from the Pareto distribution:
f
Y
(
y
) =
1
1
+
±
²
³
³
´
µ
y
k
,
y
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 Spring '08
 Monrad
 Probability

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