Uniformly most powerful test

Instatistical hypothesis testing,auniformly most powerful(UMP)testis ahypothesis testwhich has thegreatestpower$1-\beta$among all possible tests of a givensizeα.For example, according to theNeyman–Pearson lemma,thelikelihood-ratiotest is UMP for testing simple (point) hypotheses.

Setting

Let$X$denote a random vector (corresponding to the measurements), taken from aparametrized familyofprobability density functionsorprobability 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$\varphi (x)$with a reject region$R$(a subset of measurement space).

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

meaning that$H_{1}$is in force if the measurement$X\in R$and that$H_{0}$is in force if the measurement$X\in R^{c}$. Note that$R\cup R^{c}$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

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

we have

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

The Karlin–Rubin theorem

The Karlin–Rubin theorem can be regarded as an extension of the Neyman–Pearson lemma for composite hypotheses.Consider a scalar measurement having a probability density function parameterized by a scalar parameterθ,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:

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

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

is the UMP test of sizeαfor 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-dimensionalexponential familyofprobability density functionsorprobability mass functionswith

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

has a monotone non-decreasing likelihood ratio in thesufficient 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

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

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 isuniformlymost powerful in these situations.

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.