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

Let${\displaystyle X}$ denote a random vector (corresponding to the measurements), taken from aparametrized family ofprobability density functions orprobability mass functions${\displaystyle f_{\theta }(x)}$, which depends on the unknown deterministic parameter${\displaystyle \theta \in \mathrm{\Theta }}$. The parameter space${\displaystyle \mathrm{\Theta }}$ is partitioned into two disjoint sets${\displaystyle {\mathrm{\Theta }}_0}$ and${\displaystyle {\mathrm{\Theta }}_1}$. Let${\displaystyle H_0}$ denote the hypothesis that${\displaystyle \theta \in {\mathrm{\Theta }}_0}$, and let${\displaystyle H_1}$ denote the hypothesis that${\displaystyle \theta \in {\mathrm{\Theta }}_1}$.The binary test of hypotheses is performed using a test function${\displaystyle \phi (x)}$ with a reject region${\displaystyle R}$ (a subset of measurement space).

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${\displaystyle \phi (x)}$ is UMP of size${\displaystyle \alpha }$ if for any other test function${\displaystyle {\phi }^{\prime }(x)}$ satisfying

${\displaystyle \underset{\theta \in {\mathrm{\Theta }}_0}{sup}\mathrm{E}[{\phi }^{\prime }(X)|\theta ]={\alpha }^{\prime }\le \alpha =\underset{\theta \in {\mathrm{\Theta }}_0}{sup}\mathrm{E}[\phi (X)|\theta ]}$

we have

${\displaystyle \mathrm{\forall }\theta \in {\mathrm{\Theta }}_1,\mathrm{E}[{\phi }^{\prime }(X)|\theta ]=1-{\beta }^{\prime }(\theta )\le 1-\beta (\theta )=\mathrm{E}[\phi (X)|\theta ].}$

The Karlin–Rubin theorem (named forSamuel Karlin andHerman Rubin) 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 parameterθ, 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${\displaystyle {\theta }_1\ge {\theta }_0}$ (meaning that the greater${\displaystyle x}$ is, the more likely${\displaystyle H_1}$ is), then the threshold test:

where${\displaystyle x_0}$ is chosen such that${\displaystyle {\mathrm{E}}_{{\theta }_0}\phi (X)=\alpha }$

is the UMP test of sizeα for testing${\displaystyle H_0:\theta \le {\theta }_0\text{ vs. }{H}_1:\theta >{\theta }_0.}$

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-dimensionalexponential family ofprobability density functions orprobability mass functions with

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

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

Let${\displaystyle X=(X_0,\dots ,X_{M-1})}$ denote i.i.d. normally distributed${\displaystyle N}$-dimensional random vectors with mean${\displaystyle \theta m}$ and covariance matrix${\displaystyle R}$. We then have

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

is the UMP test of size${\displaystyle \alpha }$ for testing${\displaystyle H_0:\theta ⩽{\theta }_0}$ 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). As a result, no test isuniformly most powerful in these situations.

This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing 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.

• Statistical hypothesis testing

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