STAT 410
Examples for 10/26/2011
Fall 2011
Theorem 1
(
Factorization Theorem
)
:
Let
X
1
, X
2
, … , X
n
denote random variables with joint p.d.f. or p.m.f.
f
(
x
1
,
x
2
,
…
,
x
n
;
θ
)
,
which depends on the parameter
θ
.
The statistic
Y =
u
(
X
1
,
X
2
,
…
,
X
n
)
is
sufficient
for
θ
if and only if
f
(
x
1
,
x
2
,
…
,
x
n
;
θ
)
=
φ
[
u
(
x
1
,
x
2
,
…
,
x
n
)
;
θ
]
⋅
h
(
x
1
,
x
2
,
…
,
x
n
)
,
where
depends on
x
1
,
x
2
,
…
,
x
n
only through
u
(
x
1
,
x
2
,
…
,
x
n
)
and
h
(
x
1
,
x
2
,
…
,
x
n
)
does not depend on
θ
.
½
.
Let
X
1
, X
2
, … , X
n
be a random sample of size
n
from the distribution with
probability density function
( ) ( ) ( )
θ
2
X
X
ln
1
θ
θ
;
x
x
x
f
x
f
⋅

=
=
,
x
> 1,
θ
> 1.
Find a sufficient statistic
Y =
u
(
X
1
, X
2
, … , X
n
)
for
θ
.
f
(
x
1
;
θ
)
f
(
x
2
;
θ
)
…
f
(
x
n
;
θ
)
=
( )
∏
=
⋅

n
i
i
i
x
x
1
θ
2
ln
1
θ
=
( )
∏
∏
=

=
⋅
⋅

n
i
i
n
i
i
n
x
x
1
θ
1
2
ln
1
θ
.
⇒
Y
1
=
∏
=
n
i
i
1
X
is a sufficient statistic for
θ
.
⇒
Y
2
=
ln Y
1
=
ln
∏
=
n
i
i
1
X
=
∑
=
n
i
i
1
X
ln
is also a sufficient statistic for
θ
.