STAT 409
Homework #11
Fall 2009
(due Friday, November 13, by 4:00 p.m.)
1.
Let
λ
> 0
and let
X
1
, X
2
, … , X
n
be independent random variables, each with
the probability density function
f
(
x
) =
<
≥
+
1
0
1
1
λ
λ
x
x
x
.
We wish to test
H
0
:
λ
= 1
vs.
H
1
:
λ
> 1.
a)
Find a sufficient statistic for
λ
.
f
(
x
1
;
λ
)
f
(
x
2
;
λ
)
…
f
(
x
n
;
λ
)
=
1
1
λ
λ
+
=
∏
n
i
i
n
x
.
⇒
∏
=
n
i
i
x
1
is sufficient for
λ
.
b)
Find a uniformly most powerful rejection region.
That is, find a rejection region that is most powerful for testing
H
0
:
λ
= 1
vs.
H
1
:
λ
=
λ
1
for all
λ
1
> 1.
Hint:
It should look like
“Reject
H
0
if
Y
≤
c
”
or
“Reject
H
0
if
Y
≥
c
”,
where
Y =
u
(
X
1
, X
2
, … , X
n
)
is a sufficient statistic for
λ
.
( 29
,
...
,
,
2
1
λ
n
x
x
x
=
( 29
( 29
,
...
,
,
;
,
...
,
,
;
1
2
1
2
1
λ
L
L
n
n
x
x
x
x
x
x
=
1
1
2
1
1
2
2
2
2
1
λ
λ
λ
...
...
λ









n
n
n
x
x
x
x
x
x
=
n
n
x
x
x
λ
1
1
2
1
1
λ
λ
λ
...



=
1
1
λ
λ
1

=
∏
n
i
i
n
x
.
Since
λ
> 1,
( 29
,
...
,
,
2
1
λ
n
x
x
x
≤
k
⇔
∏
=
n
i
i
x
1
≤
c
.