8-4
Solutions Manual for Statistical Inference
b. The LRT statistic is
λ
(
x
) =
sup
β
(1
/β
n
)
e
-
Σ
i
x
i
/β
sup
β,γ
(
γ
n
/β
n
)(
i
x
i
)
γ
-
1
e
-
Σ
i
x
γ
i
/β
.
The numerator is maximized at
ˆ
β
0
= ¯
x
. For fixed
γ
, the denominator is maximized at
ˆ
β
γ
=
∑
i
x
γ
i
/n
. Thus
λ
(
x
) =
¯
x
-
n
e
-
n
sup
γ
(
γ
n
/
ˆ
β
n
γ
)(
i
x
i
)
γ
-
1
e
-
Σ
i
x
γ
i
/
ˆ
β
γ
=
¯
x
-
n
sup
γ
(
γ
n
/
ˆ
β
n
γ
)(
i
x
i
)
γ
-
1
.
The denominator cannot be maximized in closed form. Numeric maximization could be used
to compute the statistic for observed data
x
.
8.8 a. We will first find the MLEs of
a
and
θ
. We have
L
(
a, θ
|
x
)
=
n
i
=1
1
√
2
πaθ
e
-
(
x
i
-
θ
)
2
/
(2
aθ
)
,
log
L
(
a, θ
|
x
)
=
n
i
=1
-
1
2
log(2
πaθ
)
-
1
2
aθ
(
x
i
-
θ
)
2
.
Thus
∂
log
L
∂a
=
n
i
=1
-
1
2
a
+
1
2
θa
2
(
x
i
-
θ
)
2
=
-
n
2
a
+
1
2
θa
2
n
i
=1
(
x
i
-
θ
)
2
set
=
0
∂
log
L
∂θ
=
n
i
=1
-
1
2
θ
+
1
2
aθ
2
(
x
i
-
θ
)
2
+
1
aθ
(
x
i
-
θ
)
=
-
n
2
θ
+
1
2
aθ
2
n
i
=1
(
x
i
-
θ
)
2
+
n
¯
x
-
nθ
aθ
set
=
0
.
We have to solve these two equations simultaneously to get MLEs of
This is the end of the preview.
Sign up
to
access the rest of the document.
- Spring '12
- Dr.Hackney
- Statistics, Ratio, Mathematics in medieval Islam
-
Click to edit the document details