114
CHAPTER 2. ELEMENTS OF STATISTICAL INFERENCE
2.5
Hypothesis Testing
We assume that
Y
1
, . . . , Y
n
have a joint distribution which depends on the un-
known parameters
ϑ
= (
ϑ
1
, . . . , ϑ
p
)
T
. The set of all possible values of
ϑ
is called
the
parameter space
Θ
.
Example
2.31
.
Parameter spaces for various distributions:
1.
Y
i
∼
iid
Bin(
m, p
)
, where
m
is known,
i
= 1
,
2
, . . . , n
. Then
ϑ
=
p
and
Θ = (0
,
1)
.
2.
Y
i
∼
iid
Poisson(
λ
)
,
i
= 1
,
2
, . . . , n
. Then
ϑ
=
λ
and
Θ =
R
+
.
3.
Y
i
∼
iid
N
(
μ, σ
2
)
,
i
= 1
,
2
, . . . , n
. Then
ϑ
= (
μ, σ
2
)
T
and
Θ =
R
×
R
+
.
4.
Y
i
∼ N
(
β
0
+
β
1
x
i
, σ
2
)
independently for
i
= 1
,
2
, . . . , n
.
Then
ϑ
=
(
β
0
, β
1
, σ
2
)
T
and
Θ =
R
2
×
R
+
.
A hypothesis restricts
ϑ
to lie in
Θ
star
, where
Θ
star
⊂
Θ
. If we wish to test whether
ϑ
∈
Θ
star
, then we test the
null hypothesis
H
0
:
ϑ
∈
Θ
star
against the
alternative
hypothesis
H
1
:
ϑ
∈
Θ
\
Θ
star
.
Example
2.32
.
Examples of sets
Θ
star
restricted by null hypotheses:
1.
Y
i
∼
iid
Poisson(
λ
)
,
i
= 1
,
2
, . . . , n
. If we test
H
0
:
λ
=
λ
0
against
H
1
:
λ
negationslash
=
λ
0
then
Θ
star
=
{
λ
0
}
.
2.
Y
i
∼
iid
N
(
μ, σ
2
)
,
i
= 1
,
2
, . . . , n
. If we test
H
0
:
μ
=
μ
0
against
H
1
:
μ
negationslash
=
μ
0
then
Θ
star
=
{
μ
0
} ×
R
+
.
3.
Y
i
∼ N
(
β
0
+
β
1
x
i
, σ
2
)
independently for
i
= 1
,
2
, . . . , n
. For testing
H
0
:
β
1
= 0
against
H
1
:
β
1
negationslash
= 0
we have
Θ
star
=
R
× {
0
} ×
R
+
.
If
Θ
star
is a single point, the hypothesis is
simple
.
Otherwise, the hypothesis is
composite
.
The set of all possible values
y
of the random sample
Y
is called the
sample
space
Y
.