Indian Institute of Technology Bombay
SI 422: Regression Analysis
January 10  March 23, 2020
2
Notation
1.
A
0
denotes transpose of the matrix
A
.
2.
1
n
is a
n
×
1 vector with all entries equal to 1.
3.
0
n
is a
n
×
1 vector with all entries equal to 0.
4.
O
n
×
m
is a
n
×
m
matrix with all entries equal to 0.
5.
I
n
is the
n
×
n
identity matrix whose diagonal entries are 1 and offdiagonal entries are 0.
6. A diagonal matrix of order
p
with diagonal entries
a
1
, a
2
, . . . , a
p
is denoted by Diag
{
a
1
, a
2
, . . . , a
p
}
.
7. Ordered eigenvalues of
p
×
p
real symmetric matrix
A
is denoted by
{
λ
1
(
A
)
, λ
2
(
A
)
, . . . λ
p
(
A
)
}
.
Therefore,
λ
1
(
A
)
≥
λ
2
(
A
)
≥ · · · ≥
λ
p
(
A
) and
λ
j
(
A
) denotes the
j
th largest eigenvalue
of
A
.
8. For
p
×
p
real symmetric matrix
A
, define Λ
A
:= Diag(
λ
1
(
A
)
, λ
2
(
A
)
, . . . λ
p
(
A
)).
9.
v
j
(
A
) denotes the eigenvector of
A
corresponding to the eigenvalue
λ
j
(
A
).
10. For
p
×
p
real symmetric matrix
A
, define
V
A
:= [
v
1
(
A
)
v
2
(
A
)
· · ·
v
p
(
A
)] = ((
V
ij,
A
))
1
≤
i,j
≤
p
.
Note that
V
A
is also a
p
×
p
matrix.
11. For
p
×
p
real symmetric matrix
A
, define
V
A
,k
:= [
v
1
(
A
)
v
2
(
A
)
· · ·
v
k
(
A
)]
∀
1
≤
k
≤
p
.
Note that
V
A
is a
p
×
k
matrix.
12. Det(
A
) denotes determinant of the square matrix
A
.
13.
Z
is always a
N
(0
,
1) random variable.
14.
τ
α
is the upper
α
point or 100(1

α
)th percentile of the standard normal distribution.
15.
χ
2
ν
is a
χ
2
distributed random variable or
χ
2
distribution with degrees of freedom (DF)
ν
.
16.
χ
2
0
ν,λ
is a noncentral
χ
2
distribution with DF
ν
and noncentrality parameter
λ
.
17. (
χ
2
ν
)
α
is the upper
α
point or 100(1

α
)th percentile of the
χ
2
ν
distribution.
18.
t
ν
is a tdistributed random variable or tdistribution with degrees of freedom (DF)
ν
.
19.
t
0
ν,δ
is a noncentral tdistribution with DF
ν
and noncentrality parameter
δ
.
20. (
t
ν
)
α
is the upper
α
point or 100(1

α
)th percentile of the
t
ν
distribution.
21.
F
ν,η
is a Fdistributed random variable or Fdistribution with DF
ν, η
.
22.
F
0
ν,η,λ
is a noncentral Fdistribution with DF
ν, η
and noncentrality parameter
λ
.
23. (
F
ν,η
)
α
is the upper
α
point or 100(1

α
)th percentile of the
F
ν,η
distribution.
24. Let
X
be a random variable and
f
(
·
) be a measurable function. Then
f
(
X
)
observed
is the
observed value of
f
(
X
).
25. For a scaler
a
,
a
∼
0 indicates that
a
is close to zero.
26. For a set
A
,

A

denotes cardinality of
A
.
27. For two sets
A
and
B
,
A
×
B
=
{
(
a, b
) :
a
∈
A, b
∈
B
}
.
Chapter 0
Prerequisites
Some basic results in probability:
(BP1) If
X
∼ N
(0
,
1), then
X
2
∼
χ
2
1
.
(BP2) If
X
i
∼ N
(0
,
1) independently across
i
= 1
,
2
, . . . , n
. Then
∑
n
i
=1
X
2
i
∼
χ
2
n
.
(BP3)
E
(
χ
2
ν
) =
ν
.
(BP4) If
X
∼ N
(
δ,
1), then
X
2
∼
χ
2
0
1
,δ
2
.
(BP5) If
X
i
∼ N
(
δ
i
,
1) independently across
i
= 1
,
2
, . . . , n
. Then
n
X
i
=1
X
2
i
∼
χ
2
0
n,λ
where
λ
=
n
X
i
=1
δ
2
i
.
(BP6)
E
(
χ
2
0
ν,λ
) =
ν
+
λ
.
(BP7) Suppose
X
∼ N
(0
,
1) and
V
∼
χ
2
ν
are two independent random variables. Then
X
p
V/ν
∼
t
ν
.
(BP8)
E
(
t
ν
) = 0.
(BP9) Suppose
X
∼ N
(
δ,
1) and
V
∼
χ
2
ν
are two independent random variables. Then
X
p
V/ν
∼
t
0
ν,δ
.
(BP10)
E
(
t
0
ν,δ
) =
δ
p
ν
2
Γ
(
ν

1
2
)
Γ
(
ν
2
)
if
ν >
1.
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