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{{Short description|Hypothesis test}}
In [[Statistical hypothesis testing|statistical hypothesis testing]], a '''uniformly most powerful''' ('''UMP''') '''test''' is a [[Statistical hypothesis testing|hypothesis test]] which has the '''greatest power <math>1 - \beta</math>''' among all possible tests of a given [[Type I and type II errors|size]] ''α''. For example, according to the [[Neyman–Pearson lemma]], the [[Likelihood-ratio test|likelihood-ratio]] test is UMP for testing simple (point) hypotheses.

== Setting ==
Let <math>X</math> denote a random vector (corresponding to the measurements), taken from a parametrized family of [[Physics:Probability density function|probability density function]]s or [[Probability mass function|probability mass function]]s  <math>f_{\theta}(x)</math>, which depends on the unknown deterministic parameter <math>\theta \in \Theta</math>. The parameter space <math>\Theta</math> is partitioned into two disjoint sets <math>\Theta_0</math> and <math>\Theta_1</math>. Let <math>H_0</math> denote the hypothesis that <math>\theta \in \Theta_0</math>, and let <math>H_1</math> denote the hypothesis that <math>\theta \in \Theta_1</math>.
The binary test of hypotheses is performed using a test function <math>\varphi(x)</math> with a reject region <math>R</math> (a subset of measurement space).
:<math>\varphi(x) = 
\begin{cases}
1 & \text{if } x \in R \\
0 & \text{if } x \in R^c
\end{cases}</math>
meaning that <math>H_1</math> is in force if the measurement <math> X \in R</math> and that <math>H_0</math> is in force if the measurement <math>X\in R^c</math>.
Note that <math>R \cup R^c</math> is a disjoint covering of the measurement space.

== Formal definition ==
A test function <math>\varphi(x)</math> is UMP of size <math>\alpha</math> if for any other test function <math>\varphi'(x)</math> satisfying
:<math>\sup_{\theta\in\Theta_0}\; \operatorname{E}[\varphi'(X)|\theta]=\alpha'\leq\alpha=\sup_{\theta\in\Theta_0}\; \operatorname{E}[\varphi(X)|\theta]\,</math>
we have
:<math> \forall \theta \in \Theta_1, \quad \operatorname{E}[\varphi'(X)|\theta]= 1 - \beta'(\theta) \leq 1 - \beta(\theta) =\operatorname{E}[\varphi(X)|\theta].</math>

== The Karlin–Rubin theorem ==
The Karlin–Rubin theorem can be regarded as an extension of the Neyman–Pearson lemma for composite hypotheses.<ref>Casella, G.; Berger, R.L. (2008), ''Statistical Inference'', Brooks/Cole. {{ISBN|0-495-39187-5}} (Theorem 8.3.17)</ref> Consider a scalar measurement having a probability density function parameterized by a scalar parameter ''θ'', and define the likelihood ratio <math> l(x) = f_{\theta_1}(x) / f_{\theta_0}(x)</math>.
If <math>l(x)</math> is monotone non-decreasing, in <math>x</math>, for any pair <math>\theta_1 \geq \theta_0</math> (meaning that the greater <math>x</math> is, the more likely <math>H_1</math> is), then the threshold test:
:<math>\varphi(x) = 
\begin{cases}
1 & \text{if } x > x_0 \\
0 & \text{if } x < x_0
\end{cases}</math>
:where <math>x_0</math> is chosen such that <math>\operatorname{E}_{\theta_0}\varphi(X)=\alpha</math>

is the UMP test of size ''α'' for testing <math> H_0: \theta \leq \theta_0 \text{ vs. } H_1: \theta > \theta_0 .</math>

Note that exactly the same test is also UMP for testing <math> H_0: \theta = \theta_0 \text{ vs. } H_1: \theta > \theta_0 .</math>

== 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|exponential family]] of [[Physics:Probability density function|probability density function]]s or [[Probability mass function|probability mass function]]s with
:<math>f_\theta(x) = g(\theta) h(x) \exp(\eta(\theta) T(x))</math>
has a monotone non-decreasing likelihood ratio in the sufficient statistic <math>T(x)</math>, provided that <math>\eta(\theta)</math> is non-decreasing.

== Example ==
Let <math>X=(X_0 ,\ldots , X_{M-1})</math> denote i.i.d. normally distributed <math>N</math>-dimensional random vectors with mean <math>\theta m</math> and covariance matrix <math>R</math>. We then have

:<math>\begin{align}
f_\theta (X) = {} & (2 \pi)^{-MN/2} |R|^{-M/2} \exp \left\{-\frac 1 2 \sum_{n=0}^{M-1} (X_n - \theta m)^T R^{-1}(X_n - \theta m) \right\} \\[4pt]
= {} & (2 \pi)^{-MN/2} |R|^{-M/2} \exp \left\{-\frac 1 2 \sum_{n=0}^{M-1} \left (\theta^2 m^T R^{-1} m \right ) \right\} \\[4pt]
& \exp \left\{-\frac 1 2 \sum_{n=0}^{M-1} X_n^T R^{-1} X_n \right\} \exp \left\{\theta m^T R^{-1} \sum_{n=0}^{M-1}X_n \right\}
\end{align}</math>

which is exactly in the form of the exponential family shown in the previous section, with the sufficient statistic being

: <math>T(X) = m^T R^{-1} \sum_{n=0}^{M-1}X_n.</math>

Thus, we conclude that the test
:<math>\varphi(T) =  \begin{cases} 1 & T > t_0 \\ 0 & T < t_0 \end{cases} \qquad \operatorname{E}_{\theta_0} \varphi (T) = \alpha</math>

is the UMP test of size <math>\alpha</math> for testing <math>H_0: \theta \leqslant \theta_0</math> vs. <math>H_1: \theta > \theta_0</math>

== 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 <math>\theta_1</math> where <math>\theta_1 > \theta_0</math>) is different from the most powerful test of the same size for a different value of the parameter (e.g. for <math>\theta_2</math> where <math>\theta_2 < \theta_0</math>). As a result, no test is '''uniformly''' most powerful in these situations.


== References ==
{{reflist}}

== Further reading ==
* {{cite book|last1=Ferguson|first1=T. S.|title=Mathematical Statistics: A decision theoretic approach|publisher=Academic Press|place=New York|section=Sec. 5.2: ''Uniformly most powerful tests''|year=1967}}
* {{cite book|last1=Mood|first1=A. M.|last2=Graybill|first2=F. A.|last3=Boes|first3=D. C.|title=Introduction to the theory of statistics|edition=3rd|publisher=McGraw-Hill|place=New York|section=Sec. IX.3.2: ''Uniformly most powerful tests''|year=1974}}
* L. L. Scharf, ''Statistical Signal Processing'', Addison-Wesley, 1991, section 4.7.

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