# Uniformly most powerful test

In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power $1-\beta$ 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.

## Setting

Let $X$ denote a random vector (corresponding to the measurements), taken from a parametrized family of probability density functions or probability mass functions $f_{\theta }(x)$, which depends on the unknown deterministic parameter $\theta \in \Theta$. The parameter space $\Theta$ is partitioned into two disjoint sets $\Theta _{0}$ and $\Theta _{1}$. Let $H_{0}$ denote the hypothesis that $\theta \in \Theta _{0}$, and let $H_{1}$ denote the hypothesis that $\theta \in \Theta _{1}$. The binary test of hypotheses is performed using a test function $\varphi (x)$ with a reject region $R$ (a subset of measurement space).
    $\varphi (x)={\begin{cases}1&{\text{if }}x\in R\\0&{\text{if }}x\in R^{c}\end{cases}}$
meaning that $H_{1}$ is in force if the measurement $X\in R$ and that $H_{0}$ is in force if the measurement $X\in R^{c}$. Note that $R\cup R^{c}$ is a disjoint covering of the measurement space.

## Formal definition

A test function $\varphi (x)$ is UMP of size $\alpha$ if for any other test function $\varphi '(x)$ satisfying
    $\sup _{\theta \in \Theta _{0}}\;\operatorname {E} [\varphi '(X)|\theta ]=\alpha '\leq \alpha =\sup _{\theta \in \Theta _{0}}\;\operatorname {E} [\varphi (X)|\theta ]\,$
we have
    $\forall \theta \in \Theta _{1},\quad \operatorname {E} [\varphi '(X)|\theta ]=1-\beta '(\theta )\leq 1-\beta (\theta )=\operatorname {E} [\varphi (X)|\theta ].$

## 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 $l(x)=f_{\theta _{1}}(x)/f_{\theta _{0}}(x)$. If $l(x)$ is monotone non-decreasing, in $x$, for any pair $\theta _{1}\geq \theta _{0}$ (meaning that the greater $x$ is, the more likely $H_{1}$ is), then the threshold test:
    $\varphi (x)={\begin{cases}1&{\text{if }}x>x_{0}\\0&{\text{if }}x<x_{0}\end{cases}}$
    where $x_{0}$ is chosen such that $\operatorname {E} _{\theta _{0}}\varphi (X)=\alpha$
is the UMP test of size α for testing $H_{0}:\theta \leq \theta _{0}{\text{ vs. }}H_{1}:\theta >\theta _{0}.$
Note that exactly the same test is also UMP for testing $H_{0}:\theta =\theta _{0}{\text{ vs. }}H_{1}:\theta >\theta _{0}.$

## 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 of probability density functions or probability mass functions with
    $f_{\theta }(x)=g(\theta )h(x)\exp(\eta (\theta )T(x))$
has a monotone non-decreasing likelihood ratio in the sufficient statistic $T(x)$, provided that $\eta (\theta )$ is non-decreasing.

## Example

Let $X=(X_{0},\ldots ,X_{M-1})$ denote i.i.d. normally distributed $N$-dimensional random vectors with mean $\theta m$ and covariance matrix $R$. We then have
    ${\begin{aligned}f_{\theta }(X)=&(2\pi )^{-MN/2}|R|^{-M/2}\exp \left\{-{\frac {1}{2}}\sum _{n=0}^{M-1}(X_{n}-\theta m)^{T}R^{-1}(X_{n}-\theta m)\right\}\\[4pt]=&(2\pi )^{-MN/2}|R|^{-M/2}\exp \left\{-{\frac {1}{2}}\sum _{n=0}^{M-1}\left(\theta ^{2}m^{T}R^{-1}m\right)\right\}\\[4pt]&\exp \left\{-{\frac {1}{2}}\sum _{n=0}^{M-1}X_{n}^{T}R^{-1}X_{n}\right\}\exp \left\{\theta m^{T}R^{-1}\sum _{n=0}^{M-1}X_{n}\right\}\end{aligned}}$
which is exactly in the form of the exponential family shown in the previous section, with the sufficient statistic being
    $T(X)=m^{T}R^{-1}\sum _{n=0}^{M-1}X_{n}.$
Thus, we conclude that the test
    $\varphi (T)={\begin{cases}1&T>t_{0}\\0&T<t_{0}\end{cases}}\qquad \operatorname {E} _{\theta _{0}}\varphi (T)=\alpha$
is the UMP test of size $\alpha$ for testing $H_{0}:\theta \leqslant \theta _{0}$ vs. $H_{1}:\theta >\theta _{0}$

## Further discussion

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 $\theta _{1}$ where $\theta _{1}>\theta _{0}$) is different from the most powerful test of the same size for a different value of the parameter (e.g. for $\theta _{2}$ where $\theta _{2}<\theta _{0}$). As a result, no test is uniformly most powerful in these situations.