# 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 $\varphi (x)$ .

$\varphi (x)={\begin{cases}1&{\text{if }}\theta \in \Theta _{1}\\0&{\text{if }}\theta \in \Theta _{0}\end{cases}}$ meaning that $H_{1}$ is in force if the measurement $X\in \Theta _{1}$ and that $H_{0}$ is in force if the measurement $X\in \Theta _{0}$ . Note that $\Theta _{0}\cup \Theta _{1}$ 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

$\sup _{\theta \in \Theta _{0}}\;\operatorname {E} _{\theta }\varphi '(X)=\alpha '\leq \alpha =\sup _{\theta \in \Theta _{0}}\;\operatorname {E} _{\theta }\varphi (X)\,$ we have

$\forall \theta \in \Theta _{1},\quad \operatorname {E} _{\theta }\varphi '(X)=1-\beta '(\theta )\leq 1-\beta (\theta )=\operatorname {E} _{\theta }\varphi (X).$ ## 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 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 is the UMP test of size $\alpha$ for testing $H_{0}:\theta \leqslant \theta _{0}$ vs. $H_{1}:\theta >\theta _{0}$ ## 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 $\theta _{1}$ 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.