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.

Setting

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.

Formal definition

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

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 ${\displaystyle H_{0}:\theta =\theta _{0}{\text{ vs. }}H_{1}:\theta >\theta _{0}.}$

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

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

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

Example

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. ${\displaystyle H_{1}:\theta >\theta _{0}}$

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 ${\displaystyle \theta _{1}}$ 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.

References

1. Casella, G.; Berger, R.L. (2008), Statistical Inference, Brooks/Cole. ISBN 0-495-39187-5 (Theorem 8.3.17)

Further reading

• 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.