2024年4月1日发(作者：溥令枫)
1. Let
X
1
,X
2
,
,X
n
be a random sample from the
Gamma(
,
)
distribution
f(x|
,
)
1
(
)
x
1
e
,
x0
,
0
,
0
.
x
(a) ( 8 %) Find the method of moment estimates of
and
.
(b) ( 7 %) Find the MLE of
, assuming
is known.
(c) ( 7 %) Giving
0
, find the Cramer-Rao lower bound of estimates of
.
(d) ( 8 %) Giving
0
, find the UMVUE of
.
2. Suppose that
X
1
,X
2
,
,X
n
are iid ~
B(2,p)
,
p(0,1)
. Let
(p)2p(1p)
.
(a) ( 5 %) Show that
T
X
i
is a sufficient statistic for
p
.
i
1
n
1, if X
1
1
(b) ( 5 %) Let
Y
. Show that
Y
is an unbiased estimate of
(p)
.
0, if X
1
1
(c) (10%) Find the UMVUE
W
of
(p)
.
3. Let
X
1
, X
2
,
,X
n
be a random sample from a
Poisson(
)
,
0
, distribution.
Consider testing
H
0
:
1
vs
H
1
:
3
.
(a) (10%) Find a UMP level
test,
0
1
.
(b) ( 7 %) For
n3
, the test rejects
H
0
, if
X
1
X
2
X
3
5
.
Find the power function
(
)
of the test.
(c) ( 8 %) For
n3
, the test rejects
H
0
, if
X
1
X
2
X
3
5
.
Evaluate the size and the power of the test.
4. (10%) Let
X
1
, X
2
,
,X
n
be iid
Poisson()
distribution, and let the prior
distribution of
be a
Gamma(
,
)
distribution,
0
,
0
. Find the
posterior distribution of
.
5. Let
X
1
, X
2
,
,X
n
be a random sample from an exponential distribution with mean
,
0
.
(a) ( 5 %) Show that
T
X
i
is a sufficient statistic n for
.
i
1
n
(b) ( 5 %) Show that the Poisson family has a monotone likelihood ratio, MLR.
(c) ( 5 %) Find a UMP level
test of
H
0
:0
1
vs
H
1
:
1
by the
Karlin-Rubin Theorem shown below.
[Definition] A family of pdfs or pmfs
{g(t|
)|
}
has a monotone likelihood ratio,
g(t|
2
)
MLR, if for every
2
1
,
is a monotone function of
t
.
g(t|
1
)
[Karlin-Rubin Theorem] Suppose that
T
is a sufficient statistic for
and the pdfs or
pmfs
{g(t|
)|
}
has a non-decreasing monotone likelihood ratio.
Consider testing
H
0
:
0
vs
H
1
:
0
. A UMP level
test rejects
H
0
if and only if
Tt
0
, where
P
(Tt
0
)
.
0
1
數理統計期末考試試題答案
1. (a) Since
E(X)
0
0
(
)
1
x
1
(
1)
1
xedx
and
x
(
)
E(X)
2
(
)
2
m
1
1
(
2)
2
edx
(
1)
2
,
x
(
)
Let
m
1
and
m
2
(
1)
2
2
mmm
1
21
~
,
~
.
2
m
1
m
2
m
1
m
2
2
m
1
1
~
Furthermore,
m
1
X
,
1
2
m
2
m
1
n
i
1
n
1
X
i
2
X
2
n
i
1
(X
i
2
X)
2
n
(n
1)
2
S
,
n
2
(n1)S
~
The MME of
.and
are
,
2
nX
(n
1)S
nX
2
~
(b)
L(
|
,
~
x)
[
i
1
n
1
(
)
1
x
i
e]
x
i
1
[
(
)
]
n
i
1
n
1
(
x
i
)e
n
i
1
x
i
n
x)
nln
(
)
n
ln
(
1)
lnx
i
i
1
lnL(
|
,
~
i
1
x
1
n
n
n
1
n
1x
~
ˆ
Let
lnL(
|
,x)
x
0
x
i
i
.
2
i
1
n
i
1
n
2
n
n
2nx
~
Furthermore,
lnL(
|
,x)
lnx
i
223
23
i
1
n
2nxnx
2nx
n
ˆ
|
,
~
lnL(
x)
0
,
22332
ˆˆ
xx
ˆ
X
is the MLE of
. So,
2
2
n
2
n
n
2n
n
~
lnL(
|
,x)]
E
(
X)
(c)
E
[
223
i
232
i
1
2
CRLB =
E
[
(d) Since
E
(
2
2
2
n
lnL(
|
,
~
x)]
1
X
)
ˆ
X
is an unbiased estimate of
, and
,
2
1
2
2
ˆ
X
is the UMVUE of
.
Var()
CRLB,
n
2
n
X
[Or]
f(x|
,
)
1
(
)
(
)
Given
,
{f(x|
)}
is an exponential family in
.
I
(0,
)
(x)x
1
e
x
11
I
(0,
)
(x)x
1
exp[x(
)]
T
X
i
is a sufficient statistic for
.
i
1
n
ˆ
X
T
is an unbiased estimate of
and a function of sufficient Since
n
ˆ
X
is the UMVUE of
. statistics
T
, by Rao-Blackwell Theorem,
nn
2
x
i
2
x
i
2. (a)
f(x
1
,x
2
,,x
n
|p)
f(x
i
|p)
[
I(x)p(1
p)]
x
{0,1,2}i
i
1i
1
i
x
i
p
x
i
p
i
2
1
(1
p)
2n
[
I(x)()(1
p)]
[I(x)]()
{0,1,2}i{0,1,2}i
x
x
1
p1
p
i
i
1i
1
i
n
2
p
T(
~
x)2n
~
Let
g(T(x),p)
(
I
{0,1,2}
(
x
i
)
. By
)(1p)
and
h(x)
x
1
p
i
1
i
n
2
n
2
n
factorization theorem,
T
X
i
is a sufficient statistic for
p
.
i
1
n
2
p
x2
x
2
2
[Or]
f(x|p)
I(x)p(1
p)
I(x)(1
p)exp[xln()]
x
{0,1,2}
x
{0,1,2}
1
p
{f(x|p)}
is an exponential family
T
X
i
is a sufficient statistic.
i
1
n
2
12
1
(b)
E(Y)
1
P(X
1
1)
0
P(X
1
1)
2p(1p)
, so
Y
is an
1
p(1p)
unbiased estimate of
(p)
.
(c) If
X
1
,X
2
,
,X
n
,
nN
, are iid ~
B(2,p)
, then
T
X
i
~
B
(2
n
,
p
)
.
i
1
n
E(Y|T
t)
P(Y
1 &T
t)
P(X
1
1 &T
t)
P(T
t)P(T
t)
n
P(X
1
1 &
X
i
t
1)
i
2
n
P(T
t)
P(X
1
1)P(
X
i
t
1)
i
2
P(T
t)
2n
2
t
12n
t
1
2p(1
p)
p(1
p)
t
1
2n
t2n
t
p(1
p)
t
2(2n
2)!t!(2n
t)!t(2n
t)
,
t0,1,2,,2n
.
(t
1)!(2n
t
1)!(2n)!n(2n
1)
T(n
T)
By Rao-Blackwell Theorem,
W
E(Y|T)
is the UMVUE of
(
)e
.
2(2n
1)
3
2024年4月1日发(作者：溥令枫)
1. Let
X
1
,X
2
,
,X
n
be a random sample from the
Gamma(
,
)
distribution
f(x|
,
)
1
(
)
x
1
e
,
x0
,
0
,
0
.
x
(a) ( 8 %) Find the method of moment estimates of
and
.
(b) ( 7 %) Find the MLE of
, assuming
is known.
(c) ( 7 %) Giving
0
, find the Cramer-Rao lower bound of estimates of
.
(d) ( 8 %) Giving
0
, find the UMVUE of
.
2. Suppose that
X
1
,X
2
,
,X
n
are iid ~
B(2,p)
,
p(0,1)
. Let
(p)2p(1p)
.
(a) ( 5 %) Show that
T
X
i
is a sufficient statistic for
p
.
i
1
n
1, if X
1
1
(b) ( 5 %) Let
Y
. Show that
Y
is an unbiased estimate of
(p)
.
0, if X
1
1
(c) (10%) Find the UMVUE
W
of
(p)
.
3. Let
X
1
, X
2
,
,X
n
be a random sample from a
Poisson(
)
,
0
, distribution.
Consider testing
H
0
:
1
vs
H
1
:
3
.
(a) (10%) Find a UMP level
test,
0
1
.
(b) ( 7 %) For
n3
, the test rejects
H
0
, if
X
1
X
2
X
3
5
.
Find the power function
(
)
of the test.
(c) ( 8 %) For
n3
, the test rejects
H
0
, if
X
1
X
2
X
3
5
.
Evaluate the size and the power of the test.
4. (10%) Let
X
1
, X
2
,
,X
n
be iid
Poisson()
distribution, and let the prior
distribution of
be a
Gamma(
,
)
distribution,
0
,
0
. Find the
posterior distribution of
.
5. Let
X
1
, X
2
,
,X
n
be a random sample from an exponential distribution with mean
,
0
.
(a) ( 5 %) Show that
T
X
i
is a sufficient statistic n for
.
i
1
n
(b) ( 5 %) Show that the Poisson family has a monotone likelihood ratio, MLR.
(c) ( 5 %) Find a UMP level
test of
H
0
:0
1
vs
H
1
:
1
by the
Karlin-Rubin Theorem shown below.
[Definition] A family of pdfs or pmfs
{g(t|
)|
}
has a monotone likelihood ratio,
g(t|
2
)
MLR, if for every
2
1
,
is a monotone function of
t
.
g(t|
1
)
[Karlin-Rubin Theorem] Suppose that
T
is a sufficient statistic for
and the pdfs or
pmfs
{g(t|
)|
}
has a non-decreasing monotone likelihood ratio.
Consider testing
H
0
:
0
vs
H
1
:
0
. A UMP level
test rejects
H
0
if and only if
Tt
0
, where
P
(Tt
0
)
.
0
1
數理統計期末考試試題答案
1. (a) Since
E(X)
0
0
(
)
1
x
1
(
1)
1
xedx
and
x
(
)
E(X)
2
(
)
2
m
1
1
(
2)
2
edx
(
1)
2
,
x
(
)
Let
m
1
and
m
2
(
1)
2
2
mmm
1
21
~
,
~
.
2
m
1
m
2
m
1
m
2
2
m
1
1
~
Furthermore,
m
1
X
,
1
2
m
2
m
1
n
i
1
n
1
X
i
2
X
2
n
i
1
(X
i
2
X)
2
n
(n
1)
2
S
,
n
2
(n1)S
~
The MME of
.and
are
,
2
nX
(n
1)S
nX
2
~
(b)
L(
|
,
~
x)
[
i
1
n
1
(
)
1
x
i
e]
x
i
1
[
(
)
]
n
i
1
n
1
(
x
i
)e
n
i
1
x
i
n
x)
nln
(
)
n
ln
(
1)
lnx
i
i
1
lnL(
|
,
~
i
1
x
1
n
n
n
1
n
1x
~
ˆ
Let
lnL(
|
,x)
x
0
x
i
i
.
2
i
1
n
i
1
n
2
n
n
2nx
~
Furthermore,
lnL(
|
,x)
lnx
i
223
23
i
1
n
2nxnx
2nx
n
ˆ
|
,
~
lnL(
x)
0
,
22332
ˆˆ
xx
ˆ
X
is the MLE of
. So,
2
2
n
2
n
n
2n
n
~
lnL(
|
,x)]
E
(
X)
(c)
E
[
223
i
232
i
1
2
CRLB =
E
[
(d) Since
E
(
2
2
2
n
lnL(
|
,
~
x)]
1
X
)
ˆ
X
is an unbiased estimate of
, and
,
2
1
2
2
ˆ
X
is the UMVUE of
.
Var()
CRLB,
n
2
n
X
[Or]
f(x|
,
)
1
(
)
(
)
Given
,
{f(x|
)}
is an exponential family in
.
I
(0,
)
(x)x
1
e
x
11
I
(0,
)
(x)x
1
exp[x(
)]
T
X
i
is a sufficient statistic for
.
i
1
n
ˆ
X
T
is an unbiased estimate of
and a function of sufficient Since
n
ˆ
X
is the UMVUE of
. statistics
T
, by Rao-Blackwell Theorem,
nn
2
x
i
2
x
i
2. (a)
f(x
1
,x
2
,,x
n
|p)
f(x
i
|p)
[
I(x)p(1
p)]
x
{0,1,2}i
i
1i
1
i
x
i
p
x
i
p
i
2
1
(1
p)
2n
[
I(x)()(1
p)]
[I(x)]()
{0,1,2}i{0,1,2}i
x
x
1
p1
p
i
i
1i
1
i
n
2
p
T(
~
x)2n
~
Let
g(T(x),p)
(
I
{0,1,2}
(
x
i
)
. By
)(1p)
and
h(x)
x
1
p
i
1
i
n
2
n
2
n
factorization theorem,
T
X
i
is a sufficient statistic for
p
.
i
1
n
2
p
x2
x
2
2
[Or]
f(x|p)
I(x)p(1
p)
I(x)(1
p)exp[xln()]
x
{0,1,2}
x
{0,1,2}
1
p
{f(x|p)}
is an exponential family
T
X
i
is a sufficient statistic.
i
1
n
2
12
1
(b)
E(Y)
1
P(X
1
1)
0
P(X
1
1)
2p(1p)
, so
Y
is an
1
p(1p)
unbiased estimate of
(p)
.
(c) If
X
1
,X
2
,
,X
n
,
nN
, are iid ~
B(2,p)
, then
T
X
i
~
B
(2
n
,
p
)
.
i
1
n
E(Y|T
t)
P(Y
1 &T
t)
P(X
1
1 &T
t)
P(T
t)P(T
t)
n
P(X
1
1 &
X
i
t
1)
i
2
n
P(T
t)
P(X
1
1)P(
X
i
t
1)
i
2
P(T
t)
2n
2
t
12n
t
1
2p(1
p)
p(1
p)
t
1
2n
t2n
t
p(1
p)
t
2(2n
2)!t!(2n
t)!t(2n
t)
,
t0,1,2,,2n
.
(t
1)!(2n
t
1)!(2n)!n(2n
1)
T(n
T)
By Rao-Blackwell Theorem,
W
E(Y|T)
is the UMVUE of
(
)e
.
2(2n
1)
3