Composite Tests and the Likelihood Ratio Test
Charles B. Moss
August 3, 2010
I. Simple Tests against a Composite
A. Mathematically, we now can express the tests as testing between
H
0
:
θ
=
θ
0
against
H
1
:
θ
∈
Θ
1
, where Θ
1
is a subset of the
parameter space.
B. Given this specification, we must modify our definition of the
power of the test because the
β
value (the probability of accepting
the null hypothesis when it is false) is not unique. In this regard,
it is useful to develop the power function.
1.
Definition 9.4.1.
If the distribution of the sample
X
de
pends on a vector of parameters
θ
, we define the power func
tion of the test based on the critical region
R
by
Q
(
θ
) =
P
(
X
∈
R

θ
)
(1)
2.
Definition 9.4.2.
Let
Q
1
(
θ
) and
Q
2
(
θ
) be the power func
tions of two tests respectively.
Then we say that the first
test is uniformly better (or uniformly most powerful) than
the second in testing
H
0
:
θ
=
θ
0
against
H
1
:
θ
∈
Θ
1
if
Q
1
(
θ
0
) =
Q
2
(
θ
0
) and
Q
1
(
θ
)
≥
Q
2
(
θ
) for all
θ
∈
Θ
1
and
Q
1
(
θ
)
> Q
2
(
θ
) for at least one
θ
∈
Θ
1
.
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 Spring '10
 Staff
 Null hypothesis, Hypothesis testing, Statistical hypothesis testing, Likelihood function, Statistical power, likelihood ratio test

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