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75
This is a linear function of
θ
that decreases for
j < n/
2 and increases for
j > n/
2. If
n
is even,
2
j

n
= 0 if
j
=
n/
2. So the likelihood is constant between
x
(
n/
2)
and
x
((
n/
2)+1)
, and any
value in this interval is the MLE. Usually the midpoint of this interval is taken as the MLE. If
n
is odd, the likelihood is minimized at
ˆ
θ
=
x
((
n
+1)
/
2)
.
7.15 a. The likelihood is
L
(
μ,λ

x
) =
λ
n/
2
(2
π
)
n
Q
i
x
i
exp
±

λ
2
X
i
(
x
i

μ
)
2
μ
2
x
i
²
.
For ﬁxed
λ
, maximizing with respect to
μ
is equivalent to minimizing the sum in the expo
nential.
d
dμ
X
i
(
x
i

μ
)
2
μ
2
x
i
=
d
dμ
X
i
((
x
i
/μ
)

1)
2
x
i
=

X
i
2 ((
x
i
/μ
)

1)
x
i
x
i
μ
2
.
Setting this equal to zero is equivalent to setting
X
i
³
x
i
μ

1
´
= 0
,
and solving for
μ
yields ˆ
μ
n
= ¯
x
. Plugging in this ˆ
μ
n
and maximizing with respect to
λ
amounts to maximizing an expression of the form
λ
n/
2
e

λb
. Simple calculus yields
ˆ
λ
n
=
n
2
b
where
b
=
X
i
(
x
i

¯
x
)
2
2¯
x
2
x
i
.
Finally,
2
b
=
X
i
x
i
¯
x
2

2
X
i
1
¯
x
+
X
i
1
x
i
=

n
¯
x
+
X
i
1
x
i
=
X
i
³
1
x
i

1
¯
x
´
.
b. This is the same as Exercise 6.27b.
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.
 Spring '12
 Dr.Hackney
 Statistics

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