Chapter 10
10.1 ()
ii i i i i
Eax aEx a a
µµ
1
1
n
i
i
a
10.2
11 2 2 1 2 1 2
ˆˆ
[] , 1Ek k k k k k
θθ θθθ
10.4
2/ /
6
() 1
8
xx
hx e e
θθ
140 Mathematical Statistics, 8E
10.6
()Ex
µ
2
var( )xn
σ
2
222
() Ex n
σ
µ
µ
as n
10.8
1
1
12
1
1
() ()
11
() (1)( )
()
()
n
yx
y
yny
ny
gy ne e dx
ne e
ne
δδ
δδ
δ
10.9
1
1
1
11 1
11
() ( )
n
n
n
y
n
gy n dx y
β
β
ββ β
10.10
2
2
11
22 2 2
11
1
11
()
nn
i
i
ii
nn
ii
x
EEx
nn
nn
σµ σσ
Chapter 10 141
10.12 (a) 1n values before
n
y in 1
1
n
y
n
ways.
1
1
()
n
n
y
n
fy k
n
for ,,
n
yn k
10.13
222
ˆˆˆˆ
()var() () var()EE
θθθθθ
22
( ) since var( ) 0E
θθ θ
142 Mathematical Statistics, 8E
10.16
12
ˆˆ
var( ) 3var( )
θθ
11 2 2 1 2 1 2
ˆˆ
() 1Ea a a a a a
θθ θθθ
22
12 2
ˆˆ
var var( ) var( )
i
aa
θθ
10.17
/
1
(;)
x
fx e
θ
θθ
()Ex
θ
22
()2Ex
θ
22
σθ
( ) unbiasedEx
θ
2
var( )xn
θ
ln ln x
f
θθ
10.18
22
2
2
() , ( ) , var()
12
(2)(1)
nn n
nn n
EY EY Y
nn nn
ββ
β
Chapter 10 143
so the Cramèr-Rao inequality is not satisfied.
10.19 (a)
ln ( ) 1 ( )
()
fx fx
fx
θθ
() ln () ()
fx fx fx
θθ
ln ( ) () 0
fx fxdx
10.20
ln ( ) 1fx x
µ
µσσ
from Example 10.5
µ
10.21 (a)
12
[(1)] (1)Ewx wx w w
µ
µµ
12
σσ
10.22
22
22
12
var1 (1 )ww
nn
σσ
144 Mathematical Statistics, 8E
22
22 22
21 1 2
22 22
12 12
var 2
nn
σσ σσ
σσ σσ
10.23
22
22
12
var (1 )ww
nn
σσ
10.24 For
1
2
w
222
12 12
11 11
var 444nn nn
σσσ
Efficiency =
2
1212
2
2
12
12
4
()
11
4
nn
nn
nn
nn
σ
σ
Chapter 10 145
10.26
22
and
µ
θσθ
2
var( ) 2
x
θ
10.27
1
()
n
nn n
n
n
gy y
β
β
β
( ) 0
n
nnn n
n
n
EY ydy y
β
β
10.28
1Yx
2
1
var( ) var( )Yx
nn
θ
1
1
ZY n
1
()
11
()
ny
gy ne
δ
146 Mathematical Statistics, 8E
10.29 Continue from Exercise 10.12
11
1(1)
(1) (1) 11
21
(1) Exerxise 1.15 or
21
(1)(2)
2
nn
kk
nn
nn n n
yn yn
k
in
yy
nn
EY Y y y
kk
nn
nn
kik
nn
knnn
n
nk k
n
2612
for population
(1)
(2 1) 2 ( ) 1 2 1 unbiased
2
k
Ex Ex k
2
(1) (1)( )
var( ) 12 1 12
kknkkn
xnk n
Chapter 10 147
10.30 (a)
11
22
00
11111
( ) , ( ) , var( )
233412
Ex xdx Ex x dx x
1/12 1
var( ) 336
x
3
1
13 1 3 3 1 1 3
00
1
() ( ) 5
y
EYY b yy y y dy dy
13
11 3 39 3
var( ) , var( )
10 16 80 5 16 80
YY
13
13 1
cov( , ) 516 80
YY
10.31
ˆ
() ()Eb
θθ θ
22 22 2
ˆˆˆˆ
() ()2() ()2 ()
EEEE b
θθ θ θ θ θ θ θθ θ θ
148 Mathematical Statistics, 8E
10.32
1
(1 ) 1
ˆ
var( ) 4nn
θθ
θ
2
1
ˆ
() 2
n
En
θ
θ
for
1
θ
2
1
ˆ
( ) unbiased
E
θ
(b)
11
, 4 36, 9
36 4 nn
n
10.33
1
1
1
11
() ()
n
y
gy n fxdx
α
() 1 1
0 elsewhere
fx x
αα
=
1
1
(1 )
n
ny
α
1
for 1
0 elsewhere
y
αα
Chapter 10 149
10.35
1
()
n
nn n
n
n
gy y
β
0
n
y
β
10.36
x
is consistent estimate of the mean of any population with a finite variance. Since
θ
is the
mean and
22
σθ
if follows that
x
is consistent estimate of
θ
.
10.37 For any single observation and for
c
θ
,
2/ 2
()1PX e e
θθ
θθ
does not converge to
0, so
n
X
is not consistent for
θ
.
10.40
() as
1
n
n
EY n
n
ββ
asymptotically unbiased
From Example 10.6 (see Exercise 10.27)
22
var( ) 0 as
1 ( 2) ( 1)( 2)
n
n
Yn
nnn nn
ββ
consistent by Theorem 10.3
150 Mathematical Statistics, 8E
10.43
12 12 12
12 ()()
12
12
(, ) (1 )
xx nn xx
nn
fxx xx
θθ
12
12
ˆxx
nn
θ
12 12
ˆˆ
12
(
12
)()(1)
12
(1 )
ˆ
(,)
(, )
2
(,
1
)
nn nn
fxx nn
xx
hx x
θθ
θθ
θθ
by theorem, estimator is sufficient.
10.44 Try
0x
and
1x
10.45
1
1
(, )
nn
fx x
β
1
()
n
nn
n
n
gy y
β
11
1
1
1
(, )
n
nn n
nn
n
n
fx xY nny
y
β
β
independent of
sufficient
β