Uniformly most powerful test

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

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.^[1] 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:

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

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

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

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

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.

This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (November 2010) (Learn how and when to remove this message)

• Ferguson, T. S. (1967). "Sec. 5.2: Uniformly most powerful tests". Mathematical Statistics: A decision theoretic approach. New York: Academic Press.
• Mood, A. M.; Graybill, F. A.; Boes, D. C. (1974). "Sec. IX.3.2: Uniformly most powerful tests". Introduction to the theory of statistics (3rd ed.). New York: McGraw-Hill.
• L. L. Scharf, Statistical Signal Processing, Addison-Wesley, 1991, section 4.7.