Miscellaneous Solutions and Comments on the
Chapter 7 Exercises
7.4:
For
X
1
,...,X
n
iid
N
(
θ,
1), the likelihood function may be found by
setting
μ
=
θ
and
σ
2
= 1 in the expression for the joint density given in
Example 6.2.7 on page 277. This leads to
L
(
θ

x
) =
c
exp(

n
(¯
x

θ
)
2
/
2)
where
c
= (2
π
)

n/
2
exp
±

n
X
i
=1
(
x
i

¯
x
)
2
/
2
!
.
The loglikelihood is thus
`
(
θ
) = log
c

n
2
(¯
x

θ
)
2
(1)
which as a function of
θ
is a parabola which opens downward and achieves
its maximum over
all
real values of
θ
at
θ
= ¯
x
. In this problem we require
θ
≥
0. Let Θ =
{
θ
:
θ
≥
0
}
. The MLE is the value of
θ
∈
Θ which maximizes
`
(
θ
). If ¯
x
≥
0, then the overall maximum at
θ
= ¯
x
lies in Θ and is the MLE.
However, if ¯
x <
0, then
θ
= ¯
x
lies outside Θ. When ¯
x <
0 the loglikelihood is
a decreasing function when restricted to Θ and the maximum in Θ is achieved
at the endpoint
θ
= 0 which is the MLE in this case. (This is clear if you
draw a picture of a parabola opening downward with its peak at ¯
x <
0.) In
summary, the MLE is
ˆ
θ
= ¯
x
when ¯
x
≥
0 and
ˆ
θ
= 0 otherwise.
You can reach the same conclusion without using the particular expression
for
`
(
θ
) given in (1) above. You just need to show that
`
0
(
θ
) = 0 when
θ
= ¯
x
with
`
0
(
θ
)
>
0 for
θ <
¯
x
and
`
0
(
θ
)
<
0 for
θ >
¯
x
. Then if ¯
x <
0, you have
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 Fall '10
 Frade
 Probability theory, unbiased estimator, Open Set Condition

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