Dr. Hackney STA Solutions pg 125 - 8-4 Solutions Manual for...

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