Hypothesis test
In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power
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
denote a random vector (corresponding to the measurements), taken from a parametrized family of probability density functions or probability mass functions
, which depends on the unknown deterministic parameter
. The parameter space
is partitioned into two disjoint sets
and
. Let
denote the hypothesis that
, and let
denote the hypothesis that
. The binary test of hypotheses is performed using a test function
with a reject region
(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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2065416942135d82a16145ba39b6c9c71f0aa0ed)
meaning that
is in force if the measurement
and that
is in force if the measurement
. Note that
is a disjoint covering of the measurement space.
Formal definition
A test function
is UMP of size
if for any other test function
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 ]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fd1e22e4e3219ee1be9b4b74cae308886375ede)
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 ].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc8de9bd5d17381b3015cc1732c4545582976e8)
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
. If
is monotone non-decreasing, in
, for any pair
(meaning that the greater
is, the more likely
is), then the threshold test:
![{\displaystyle \varphi (x)={\begin{cases}1&{\text{if }}x>x_{0}\\0&{\text{if }}x<x_{0}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f72cf30c7d036963fac32611d30bf16ae4a10e52)
- where
is chosen such that ![{\displaystyle \operatorname {E} _{\theta _{0}}\varphi (X)=\alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/32c7607c915f00730bd752ae5470d358e2b48210)
is the UMP test of size α for testing
Note that exactly the same test is also UMP for testing
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3fd8d57722701abe943e1b3865727a6c7d195a0)
has a monotone non-decreasing likelihood ratio in the sufficient statistic
, provided that
is non-decreasing.
Example
Let
denote i.i.d. normally distributed
-dimensional random vectors with mean
and covariance matrix
. 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd4c3432fa99b15908c8849ef2f21c1a0d05741)
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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b40f6f957b1067aa8ec70eda986695124d06793)
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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c19ee286adf0d3dbc06c8bdef5b4155e73a0943b)
is the UMP test of size
for testing
vs.
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
where
) is different from the most powerful test of the same size for a different value of the parameter (e.g. for
where
). As a result, no test is uniformly most powerful in these situations.
References
- ^ 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.
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