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
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M
M
12
1
22
pp
pp
pn
ss
xt
p
np
αα
μ
−−
⎛⎞
−≤
≤
+
⎜⎟
⎝⎠
n
Chapter 8:
Principal Component Analysis
A principal component analysis (PCA) is concerned with
explaining the variance
covariance structure
of a set of variables (e.g. p) through
a few linear combinations
(e.g.
k with k < p) of these variables.
Objectives: 1. Data reduction 2. Interpretation
Analyses of principle components (PCs) are often served as intermediate steps in a much
larger investigations.
For example, the PCs may be inputs to a multiple regression, or
cluster analysis or used in factor analysis.
We will first discuss the properties of population PCs and then see how to use sample
PCs to summarize sample variation.
Population Principal Components
Algebraically, PCs are particular linear combinations of the p random variables
1
,...,
p
X
X
.
Geometrically, these linear combinations represent the selection of a new coordinate
system obtained by rotating the original system with
1
,...,
p
X
X
as the coordinate axes.
PCs depend only on the variancecovariance matrix
Σ
(or correlation matrix
ρ
).
Suppose the random vector
1
(
,...,
)'
p
X
X
=
X
have variancecovariance matrix
. Let
be p unitlength constant vectors.
Consider the linear
combinations:
.
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 Spring '10
 Hua
 Statistics, Standard Deviation, Variance, Probability theory, Singular value decomposition

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