In
statistical hypothesis testing, a uniformly most powerful (UMP) test is a
hypothesis test
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...
which has the greatest
power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
among all possible tests of a given
size
Size in general is the magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to linear dimensions ( length, width, height, diameter, perimeter), area, or volume. Size can also be m ...
''α''. For example, according to the
Neyman–Pearson lemma
In statistics, the Neyman–Pearson lemma was introduced by Jerzy Neyman and Egon Pearson in a paper in 1933. The Neyman-Pearson lemma is part of the Neyman-Pearson theory of statistical testing, which introduced concepts like errors of the seco ...
, the
likelihood-ratio test is UMP for testing simple (point) hypotheses.
Setting
Let
denote a random vector (corresponding to the measurements), taken from a
parametrized family
In mathematics and its applications, a parametric family or a parameterized family is a family of objects (a set of related objects) whose differences depend only on the chosen values for a set of parameters.
Common examples are parametrized (fa ...
of
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
s or
probability mass functions
, which depends on the unknown deterministic parameter
. The parameter space
is partitioned into two disjoint sets
and
. Let
denote the hypothesis that
, and let
denote the hypothesis that
.
The binary test of hypotheses is performed using a test function
with a reject region
(a subset of measurement space).
:
meaning that
is in force if the measurement
and that
is in force if the measurement
.
Note that
is a disjoint covering of the measurement space.
Formal definition
A test function
is UMP of size
if for any other test function
satisfying
:
we have
:
The Karlin–Rubin theorem
The Karlin–Rubin theorem can be regarded as an extension of the Neyman–Pearson lemma for composite hypotheses.
[Casella, G.; Berger, R.L. (2008), ''Statistical Inference'', Brooks/Cole. (Theorem 8.3.17)] Consider a scalar measurement having a probability density function parameterized by a scalar parameter ''θ'', and define the likelihood ratio
.
If
is monotone non-decreasing, in
, for any pair
(meaning that the greater
is, the more likely
is), then the threshold test:
:
:where
is chosen such that
is the UMP test of size ''α'' for testing
Note that exactly the same test is also UMP for testing
Important case: exponential family
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
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate ...
of
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
s or
probability mass functions with
:
has a monotone non-decreasing likelihood ratio in the
sufficient statistic , provided that
is non-decreasing.
Example
Let
denote i.i.d. normally distributed
-dimensional random vectors with mean
and covariance matrix
. We then have
:
which is exactly in the form of the exponential family shown in the previous section, with the sufficient statistic being
:
Thus, we conclude that the test
:
is the UMP test of size
for testing
vs.
Further discussion
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
where
) is different from the most powerful test of the same size for a different value of the parameter (e.g. for
where
). As a result, no test is uniformly most powerful in these situations.
References
Further reading
*
*
* L. L. Scharf, ''Statistical Signal Processing'', Addison-Wesley, 1991, section 4.7.
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Statistical hypothesis testing