Uniformly Most Powerful Test

For example, according to the Neyman–Pearson lemma, the likelihood-ratio test is UMP for testing simple (point) hypotheses.

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

${\displaystyle \varphi (x)={\begin{cases}1&{\text{if }}x\in R\\0&{\text{if }}x\in R^{c}\end{cases}}}$ 

meaning that ${\displaystyle H_{1}}$  is in force if the measurement ${\displaystyle X\in R}$  and that ${\displaystyle H_{0}}$  is in force if the measurement ${\displaystyle X\in R^{c}}$ . Note that ${\displaystyle R\cup R^{c}}$  is a disjoint covering of the measurement space.

A test function ${\displaystyle \varphi (x)}$  is UMP of size ${\displaystyle \alpha }$  if for any other test function ${\displaystyle \varphi '(x)}$  satisfying

${\displaystyle \sup _{\theta \in \Theta _{0}}\;\operatorname {E} [\varphi '(X)|\theta ]=\alpha '\leq \alpha =\sup _{\theta \in \Theta _{0}}\;\operatorname {E} [\varphi (X)|\theta ]\,}$ 

we have

${\displaystyle \forall \theta \in \Theta _{1},\quad \operatorname {E} [\varphi '(X)|\theta ]=1-\beta '(\theta )\leq 1-\beta (\theta )=\operatorname {E} [\varphi (X)|\theta ].}$ 

The Karlin–Rubin theorem 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 ${\displaystyle l(x)=f_{\theta _{1}}(x)/f_{\theta _{0}}(x)}$ . If ${\displaystyle l(x)}$  is monotone non-decreasing, in ${\displaystyle x}$ , for any pair ${\displaystyle \theta _{1}\geq \theta _{0}}$  (meaning that the greater ${\displaystyle x}$  is, the more likely ${\displaystyle H_{1}}$  is), then the threshold test:

${\displaystyle \varphi (x)={\begin{cases}1&{\text{if }}x>x_{0}\\0&{\text{if }}x$ 
where ${\displaystyle x_{0}}$  is chosen such that ${\displaystyle \operatorname {E} _{\theta _{0}}\varphi (X)=\alpha }$ 

is the UMP test of size α for testing ${\displaystyle H_{0}:\theta \leq \theta _{0}{\text{ vs. }}H_{1}:\theta >\theta _{0}.}$ 

Note that exactly the same test is also UMP for testing ${\displaystyle H_{0}:\theta =\theta _{0}{\text{ vs. }}H_{1}:\theta >\theta _{0}.}$ 

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

${\displaystyle f_{\theta }(x)=g(\theta )h(x)\exp(\eta (\theta )T(x))}$ 

has a monotone non-decreasing likelihood ratio in the sufficient statistic ${\displaystyle T(x)}$ , provided that ${\displaystyle \eta (\theta )}$  is non-decreasing.

Let ${\displaystyle X=(X_{0},\ldots ,X_{M-1})}$  denote i.i.d. normally distributed ${\displaystyle N}$ -dimensional random vectors with mean ${\displaystyle \theta m}$  and covariance matrix ${\displaystyle R}$ . We then have

${\displaystyle {\begin{aligned}f_{\theta }(X)={}&(2\pi )^{-MN/2}|R|^{-M/2}\exp \left\{-{\frac {1}{2}}\sum _{n=0}^{M-1}(X_{n}-\theta m)^{T}R^{-1}(X_{n}-\theta m)\right\}\\[4pt]={}&(2\pi )^{-MN/2}|R|^{-M/2}\exp \left\{-{\frac {1}{2}}\sum _{n=0}^{M-1}\left(\theta ^{2}m^{T}R^{-1}m\right)\right\}\\[4pt]&\exp \left\{-{\frac {1}{2}}\sum _{n=0}^{M-1}X_{n}^{T}R^{-1}X_{n}\right\}\exp \left\{\theta m^{T}R^{-1}\sum _{n=0}^{M-1}X_{n}\right\}\end{aligned}}}$ 

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

${\displaystyle T(X)=m^{T}R^{-1}\sum _{n=0}^{M-1}X_{n}.}$ 

Thus, we conclude that the test

${\displaystyle \varphi (T)={\begin{cases}1&T>t_{0}\\0&T$ 

is the UMP test of size ${\displaystyle \alpha }$  for testing ${\displaystyle H_{0}:\theta \leqslant \theta _{0}}$  vs. ${\displaystyle H_{1}:\theta >\theta _{0}}$ 

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