Uniformly most powerful test

In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power $\beta$$\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$$X$ denote a random vector (corresponding to the measurements), taken from a parametrized family of probability density functions or probability mass functions $f_{\theta }(x)$$f_{\theta}(x)$, which depends on the unknown deterministic parameter $\theta \in \Theta$$\theta \in \Theta$. The parameter space $\Theta$$\Theta$ is partitioned into two disjoint sets $\Theta _{0}$$\Theta_0$ and $\Theta _{1}$$\Theta_1$. Let $H_{0}$$H_{0}$ denote the hypothesis that $\theta \in \Theta _{0}$$\theta \in \Theta_0$, and let $H_{1}$$H_{1}$ denote the hypothesis that $\theta \in \Theta _{1}$$\theta \in \Theta_1$. The binary test of hypotheses is performed using a test function $\phi (x)$$\phi (x)$.

$\phi (x)={\begin{cases}1&{\text{if }}x\in R\\0&{\text{if }}x\in A\end{cases}}$$\phi(x) = \begin{cases} 1 & \text{if } x \in R \\ 0 & \text{if } x \in A \end{cases}$

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

Formal definition

A test function $\phi (x)$$\phi (x)$ is UMP of size $\alpha$$\alpha$ if for any other test function $\phi '(x)$$\phi'(x)$ satisfying

$\sup _{\theta \in \Theta _{0}}\;\operatorname {E} _{\theta }\phi '(X)=\alpha '\leq \alpha =\sup _{\theta \in \Theta _{0}}\;\operatorname {E} _{\theta }\phi (X)\,$$\sup_{\theta\in\Theta_0}\; \operatorname{E}_\theta\phi'(X)=\alpha'\leq\alpha=\sup_{\theta\in\Theta_0}\; \operatorname{E}_\theta\phi(X)\,$

we have

$\forall \theta \in \Theta _{1},\quad \operatorname {E} _{\theta }\phi '(X)=1-\beta '\leq 1-\beta =\operatorname {E} _{\theta }\phi (X).$$\forall \theta \in \Theta_1, \quad \operatorname{E}_\theta\phi'(X)=1-\beta'\leq 1-\beta=\operatorname{E}_\theta\phi(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)$$l(x) = f_{\theta_1}(x) / f_{\theta_0}(x)$. If $l(x)$$l(x)$ is monotone non-decreasing, in $x$$x$, for any pair $\theta _{1}\geq \theta _{0}$$\theta_1 \geq \theta_0$ (meaning that the greater $x$$x$ is, the more likely $H_{1}$$H_{1}$ is), then the threshold test:

$\phi (x)={\begin{cases}1&{\text{if }}x>x_{0}\\0&{\text{if }}x<x_{0}\end{cases}}$$\phi(x) = \begin{cases} 1 & \text{if } x > x_0 \\ 0 & \text{if } x < x_0 \end{cases}$

where $x_{0}$$x_{0}$ is chosen such that $\operatorname {E} _{\theta _{0}}\phi (X)=\alpha$$\operatorname{E}_{\theta_0}\phi(X)=\alpha$

is the UMP test of size α for testing $H_{0}:\theta \leq \theta _{0}{\text{ vs. }}H_{1}:\theta >\theta _{0}.$$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}.$$H_0: \theta = \theta_0 \text{ vs. } H_1: \theta > \theta_0 .$

Important case: The 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)=c(\theta )h(x)\exp(\pi (\theta )T(x))$$f_\theta(x) = c(\theta)h(x)\exp(\pi(\theta)T(x))$

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

Example

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

${\begin{aligned}f_{\theta }(X)&=(2\pi )^{-{\frac {MN}{2}}}|R|^{-{\frac {M}{2}}}\exp \left\{-{\frac {1}{2}}\sum _{n=0}^{M-1}(X_{n}-\theta m)^{T}R^{-1}(X_{n}-\theta m)\right\}\\&=(2\pi )^{-{\frac {MN}{2}}}|R|^{-{\frac {M}{2}}}\exp \left\{-{\frac {1}{2}}\sum _{n=0}^{M-1}\left(\theta ^{2}m^{T}R^{-1}m\right)\right\}\\&\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}}$${\begin{aligned}f_{\theta }(X)&=(2\pi )^{-{\frac {MN}{2}}}|R|^{-{\frac {M}{2}}}\exp \left\{-{\frac {1}{2}}\sum _{n=0}^{M-1}(X_{n}-\theta m)^{T}R^{-1}(X_{n}-\theta m)\right\}\\&=(2\pi )^{-{\frac {MN}{2}}}|R|^{-{\frac {M}{2}}}\exp \left\{-{\frac {1}{2}}\sum _{n=0}^{M-1}\left(\theta ^{2}m^{T}R^{-1}m\right)\right\}\\&\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}.$$T(X) = m^T R^{-1} \sum_{n=0}^{M-1}X_n.$

Thus, we conclude that the test

$\phi (T)={\begin{cases}1&T>t_{0}\\0&T<t_{0}\end{cases}}\qquad \operatorname {E} _{\theta _{0}}\phi (T)=\alpha$$\phi (T)={\begin{cases}1&T>t_{0}\\0&T<t_{0}\end{cases}}\qquad \operatorname {E} _{\theta _{0}}\phi (T)=\alpha$

is the UMP test of size $\alpha$$\alpha$ for testing $H_{0}:\theta \leqslant \theta _{0}$$H_{0}:\theta \leqslant \theta _{0}$ vs. $H_{1}:\theta >\theta _{0}$$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}$$\theta _{1}$ where $\theta _{1}>\theta _{0}$$\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}$$\theta _{2}$ where $\theta _{2}<\theta _{0}$$\theta_2 < \theta_0$). As a result, no test is uniformly most powerful in these situations.

Further reading

• L. L. Scharf, Statistical Signal Processing, Addison-Wesley, 1991, section 4.7.