Stat 430/Math 468 – Notes #11
Chapters 5, 8
Chapter 5: Inference about a Mean Vector (Continued)
Simultaneous Confidence Interval for
'
a
μ
(i.e. for
1
p
ii
i
a
μ
=
∑
)
Result:
Let
be random sample from a normal population
. The sample
mean and sample variancecovariance matrix are, respectively,
1
,...,
n
XX
(, )
p
N
μΣ
1
12
(
...
)
n
n
=+
+
+
X
X
and
1
1
1
()
n
jj
n
j
−
=
=−
−
∑
SX
X
X
'
X
.
1.
(
confidence interval)
Simultaneously
for all
2
T
1
( ,.
..,
)'
p
aa
=
a
, the
100(1
)%
α
−
confidence interval for
'
a
μ
is
,
(1
)
'
'(
pn p
pn
F
np
n
−
−
±
−
aSa
aX
)
.
Example: For choices
1
1, 0,
,0
′
=
a
K
,
′
2
0,1,
,0
=
K
a
,
K
, and
, we
have
0,0,
,1
p
′
=
a
K
11
'
=
a
μ
,
22
'
=
μ
a
,
K
, and
'
pp
=
a
μ
. The

intervals for the above choices
are
2
T
11
11
1,
1
ss
xF
n
n
αμ
−−
−≤
≤
+
22
22
2,
2
n
n
≤
+
M
M
M
,2
,
pp
pp
n
p
n
p
n
n
≤
+
2.
(
Bonferroni confidence interval
) The
)%
−
simultaneous confidence
interval for
m
linear combinations of the mean vector
is
'
, '
,..., '
m
a
μ
a
μ
a
μ
1
'
'
,
1,2,.
..,
2
in
ti
mn
−
⎛⎞
±=
⎜⎟
⎝⎠
aS
a
m
.
Example: For the restricted set
1
′
=
a
K
,
′
2
,0
=
K
)
a
,
K
, and
, where m=p, the Bonferroni confidence intervals are
(
0,0,
,1
m
′
=
a
K
11
11
1
nn
xt
p
αα
≤
+
n
22
22
21
2
p
≤
+
n
1