NTHU MATH 2820, 2008, Lecture Notes
Ch9, p.1
Hypothesis testing
• What is hypothesis testing?
Question 7.1
(What is a hypothesis testing question?)
1.
observed data.
x
1
, . . . , x
n
2.
statistical modeling.
Regard
x
1
, . . . , x
n
as a realization of random
variables
X
1
, . . . , X
n
, and assign
X
1
, . . . , X
n
a joint distribution:
a joint cdf
F
(
·|
Θ
)
,
or a joint pdf
f
(
·|
Θ
)
,
or a joint pmf
p
(
·|
Θ
)
,
where
Θ
= (
θ
1
, . . .,
θ
k
)
∈
Ω
, and
θ
0
i
s
are
fi
xed constants
, but their
values are
unknown
.
3.
point estimation.
What is the value of
Θ
?
Find a function of
X
1
, . . . , X
n
,
Θ
, to estimate
Θ
or a function of
Θ
.
4.
hypothesis testing.
Separate
Ω
into two sets
Ω
0
and
Ω
A
, where
Ω
0
∩
Ω
A
=
∅
and
Ω
0
∪
Ω
A
=
Ω
.
Use the data
X
1
, . . . , X
n
to answer the question:
which of the two
hypotheses,
H
0
:
Θ
∈
Ω
0
versus
H
A
:
Θ
∈
Ω
A
is more favorable.
made by Shao-Wei Cheng (NTHU, Taiwan)
Ch9, p.2
Definition 7.1
(
null and alternative hypotheses, simple and composite hypotheses, TBp.331,332,334
)
Question 7.3
(asymmetry between
H
0
and
H
A
)
Question 7.2
Is there a di
ff
erence between the roles of
H
0
and
H
A
?
Can we arbitrary
exchange the two hypotheses?
Can we obtain an estimate of
Θ
and accept
H
0
if
ˆ
Θ
∈
Ω
0
and reject
H
0
if
ˆ
Θ
∈
Ω
A
?
Hint
. Consider the case:
X
∼
Binomial(
n, p
)
,
H
0
:
p
= 0
.
5
v.s.
H
1
:
p
6
= 0
.
5
,
What if
n
= 10
5
and ˆ
p
= 0
.
50001?
•
H
0
is called
null hypothesis
.
•
H
A
(sometimes denoted as
H
1
) is called
alternative hypothesis
.
•
An hypothesis is said to be a
simple hypothesis
if that hypothesis
uniquely speci
fi
es the distribution.