Uniformly most powerful test
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In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power 
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.
Contents |
[edit] Setting
Let 
denote a random vector (corresponding to the measurements), taken from a parametrized family of probability density functions 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 
.
meaning that is in force if the measurement 
and that is in force if the measurement 
.
is a disjoint covering of the measurement space.
[edit] Formal definition
A test function is UMP of size 
if for any other test function 
we have:
[edit] The Karlin-Rubin theorem
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 
.
If 
is monotone non-decreasing for any pair 
(meaning that the greater 
is, the more likely is), then the threshold test:
is the UMP test of size α for testing 
Note that exactly the same test is also UMP for testing 
[edit] Important case: The exponential family
Although the Karlin-Rubin 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 
has a monotone non-decreasing likelihood ratio in the sufficient statistic T(x), provided that 
is non-decreasing.
[edit] 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. 
[edit] 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). Why is it so?
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 than 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.
[edit] References
- L. L. Scharf, Statistical Signal Processing, Addison-Wesley, 1991, section 4.7.
| This page uses content from the English-language version of Wikipedia. The original article was at Uniformly most powerful test. The list of authors can be seen in the page history. As with Psychology Wiki, the text of Wikipedia is available under the GNU Free Documentation License. |
