In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power
β
among all possible tests of a given size α. For example, according to the Neyman–Pearson lemma, the likelihood-ratio test is UMP for testing simple (point) hypotheses.
Let
X
denote a random vector (corresponding to the measurements), taken from a parametrized family of probability density functions or probability mass functions
f
θ
(
x
)
, which depends on the unknown deterministic parameter
θ
∈
Θ
. The parameter space
Θ
is partitioned into two disjoint sets
Θ
0
and
Θ
1
. Let
H
0
denote the hypothesis that
θ
∈
Θ
0
, and let
H
1
denote the hypothesis that
θ
∈
Θ
1
. The binary test of hypotheses is performed using a test function
ϕ
(
x
)
.
ϕ
(
x
)
=
{
1
if
x
∈
R
0
if
x
∈
A
meaning that
H
1
is in force if the measurement
X
∈
R
and that
H
0
is in force if the measurement
X
∈
A
. Note that
A
∪
R
is a disjoint covering of the measurement space.
A test function
ϕ
(
x
)
is UMP of size
α
if for any other test function
ϕ
′
(
x
)
satisfying
sup
θ
∈
Θ
0
E
θ
ϕ
′
(
X
)
=
α
′
≤
α
=
sup
θ
∈
Θ
0
E
θ
ϕ
(
X
)
we have
∀
θ
∈
Θ
1
,
E
θ
ϕ
′
(
X
)
=
1
−
β
′
≤
1
−
β
=
E
θ
ϕ
(
X
)
.
The Karlin–Rubin theorem can be regarded as an extension of the Neyman–Pearson lemma for composite hypotheses. Consider a scalar measurement having a probability density function parameterized by a scalar parameter θ, and define the likelihood ratio
l
(
x
)
=
f
θ
1
(
x
)
/
f
θ
0
(
x
)
. If
l
(
x
)
is monotone non-decreasing, in
x
, for any pair
θ
1
≥
θ
0
(meaning that the greater
x
is, the more likely
H
1
is), then the threshold test:
ϕ
(
x
)
=
{
1
if
x
>
x
0
0
if
x
<
x
0
where
x
0
is chosen such that
E
θ
0
ϕ
(
X
)
=
α
is the UMP test of size α for testing
H
0
:
θ
≤
θ
0
vs.
H
1
:
θ
>
θ
0
.
Note that exactly the same test is also UMP for testing
H
0
:
θ
=
θ
0
vs.
H
1
:
θ
>
θ
0
.
Although the Karlin-Rubin theorem may seem weak because of its restriction to scalar parameter and scalar measurement, it turns out that there exist a host of problems for which the theorem holds. In particular, the one-dimensional exponential family of probability density functions or probability mass functions with
f
θ
(
x
)
=
g
(
θ
)
h
(
x
)
exp
(
η
(
θ
)
T
(
x
)
)
has a monotone non-decreasing likelihood ratio in the sufficient statistic
T
(
x
)
, provided that
η
(
θ
)
is non-decreasing.
Let
X
=
(
X
0
,
…
,
X
M
−
1
)
denote i.i.d. normally distributed
N
-dimensional random vectors with mean
θ
m
and covariance matrix
R
. We then have
f
θ
(
X
)
=
(
2
π
)
−
M
N
2
|
R
|
−
M
2
exp
{
−
1
2
∑
n
=
0
M
−
1
(
X
n
−
θ
m
)
T
R
−
1
(
X
n
−
θ
m
)
}
=
(
2
π
)
−
M
N
2
|
R
|
−
M
2
exp
{
−
1
2
∑
n
=
0
M
−
1
(
θ
2
m
T
R
−
1
m
)
}
exp
{
−
1
2
∑
n
=
0
M
−
1
X
n
T
R
−
1
X
n
}
exp
{
θ
m
T
R
−
1
∑
n
=
0
M
−
1
X
n
}
which is exactly in the form of the exponential family shown in the previous section, with the sufficient statistic being
T
(
X
)
=
m
T
R
−
1
∑
n
=
0
M
−
1
X
n
.
Thus, we conclude that the test
ϕ
(
T
)
=
{
1
T
>
t
0
0
T
<
t
0
E
θ
0
ϕ
(
T
)
=
α
is the UMP test of size
α
for testing
H
0
:
θ
⩽
θ
0
vs.
H
1
:
θ
>
θ
0
Finally, we note that in general, UMP tests do not exist for vector parameters or for two-sided tests (a test in which one hypothesis lies on both sides of the alternative). The reason is that in these situations, the most powerful test of a given size for one possible value of the parameter (e.g. for
θ
1
where
θ
1
>
θ
0
) is different from the most powerful test of the same size for a different value of the parameter (e.g. for
θ
2
where
θ
2
<
θ
0
). As a result, no test is uniformly most powerful in these situations.
