62
Solutions Manual for Statistical Inference
6.6 The joint pdf is given by
f
(
x
1
, . . . , x
n

α, β
) =
n
i
=1
1
Γ(
α
)
β
α
x
i
α

1
e

x
i
/β
=
1
Γ(
α
)
β
α
n
n
i
=1
x
i
α

1
e

Σ
i
x
i
/β
.
By the Factorization Theorem, (
n
i
=1
X
i
,
∑
n
i
=1
X
i
) is sufficient for (
α, β
).
6.7 Let
x
(1)
= min
i
{
x
1
, . . . , x
n
}
,
x
(
n
)
= max
i
{
x
1
, . . . , x
n
}
,
y
(1)
= min
i
{
y
1
, . . . , y
n
}
and
y
(
n
)
=
max
i
{
y
1
, . . . , y
n
}
. Then the joint pdf is
f
(
x
,
y

θ
)
=
n
i
=1
1
(
θ
3

θ
1
)(
θ
4

θ
2
)
I
(
θ
1
,θ
3
)
(
x
i
)
I
(
θ
2
,θ
4
)
(
y
i
)
=
1
(
θ
3

θ
1
)(
θ
4

θ
2
)
n
I
(
θ
1
,
∞
)
(
x
(1)
)
I
(
∞
,θ
3
)
(
x
(
n
)
)
I
(
θ
2
,
∞
)
(
y
(1)
)
I
(
∞
,θ
4
)
(
y
(
n
)
)
g
(
T
(
x
)

θ
)
·
1
h
(
x
)
.
By the Factorization Theorem,
(
X
(1)
, X
(
n
)
, Y
(1)
, Y
(
n
)
)
is sufficient for (
θ
1
, θ
2
, θ
3
, θ
4
).
6.9 Use Theorem 6.2.13.
a.
f
(
x

θ
)
f
(
y

θ
)
=
(2
π
)

n/
2
e

Σ
i
(
x
i

θ
)
2
/
2
(2
π
)

n/
2
e

Σ
i
(
y
i

θ
)
2
/
2
= exp

1
2
n
i
=1
x
2
i

n
i
=1
y
2
i
+2
θn
(¯
y

¯
x
)
.
This is constant as a function of
θ
if and only if ¯
y
= ¯
x
; therefore
¯
X
is a minimal sufficient
statistic for
θ
.
b. Note, for
X
∼
location exponential(
θ
), the range depends on the parameter. Now
f
(
x

θ
)
f
(
y

θ
)
=
n
i
=1
(
e

(
x
i

θ
)
I
(
θ,
∞
)
(
x
i
)
)
n
i
=1
(
e

(
y
i

θ
)
I
(
θ,
∞
)
(
y
i
)
)
=
e
nθ
e

Σ
i
x
i
n
i
=1
I
(
θ,
∞
)
(
x
i
)
e
nθ
e

Σ
i
y
i
n
i
=1
I
(
θ,
∞
)
(
y
i
)
=
e

Σ
i
x
i
I
(
θ,
∞
)
(min
x
i
)
e

Σ
i
y
i
I
(
θ,
∞
)
(min
y
i
)
.
To make the ratio independent of
θ
we need the ratio of indicator functions independent
of
θ
. This will be the case if and only if min
{
x
1
, . . . , x
n
}
= min
{
y
1
, . . . , y
n
}
. So
T
(
X
) =
min
{
X
1
, . . . , X
n
}
is a minimal sufficient statistic.
c.
f
(
x

θ
)
f
(
y

θ
)
=
e

Σ
i
(
x
i

θ
)
n
i
=1
(
1 +
e

(
x
i

θ
)
)
2
n
i
=1
(
1 +
e

(
y
i

θ
)
)
2
e

Σ
i
(
y
i

θ
)
=
e

Σ
i
(
y
i

x
i
)
n
i
=1
(
1 +
e

(
y
i

θ
)
)
n
i
=1
(
1 +
e

(
x
i

θ
)
)
2
.
This is constant as a function of
θ
if and only if
x
and
y
have the same order statistics.