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
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document1.
Let
X
1
, X
2
, … , X
n
be a random sample of size
n
from a Poisson distribution
with mean
λ
.
That is,
f
(
k
;
λ
) = P
(
X
1
=
k
) =
!
λ
λ
k
e
k

⋅
,
k
= 0, 1, 2, 3, … .
a)
Use Factorization Theorem to find
Y =
u
(
X
1
,
X
2
,
…
,
X
n
),
a sufficient statistic
for
λ
.
f
(
x
1
;
λ
)
f
(
x
2
;
λ
)
…
f
(
x
n
;
λ
)
=
∏
=

⋅
n
i
i
x
x
e
i
1
!
λ
λ
=
∏
∑
=

⋅
⋅
=
n
i
i
n
x
x
e
n
i
i
1
!
λ
1
1
λ
.
By Factorization Theorem,
Y =
∑
=
n
i
i
1
X
is a sufficient statistic for
λ
.
[
⇒
X
is also a sufficient statistic for
λ
.
]
b)
Show that
P
(
X
1
=
x
1
, X
2
=
x
2
, … , X
n
=
x
n

Y =
y
)
does not depend on
λ
.
Since
Y =
∑
=
n
i
i
1
X
has a Poisson distribution with mean
n
λ
,
if
∑
=
n
i
i
x
1
=
y
,
P
(
X
1
=
x
1
, X
2
=
x
2
, … , X
n
=
x
n

Y =
y
)
=
=
( )
!
λ
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Monrad
 Statistics, Probability, Probability theory, Max, Sufficient statistic

Click to edit the document details