Uniformly most powerful test

Uniformly most powerful test

In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power 1-eta 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 X denote a random vector (corresponding to the measurements), taken from a parametrized family of probability density functions or probability mass functions f_{ heta}(x), which depends on the unknown deterministic parameter heta in Theta. The parameter space Theta is partitioned into two disjoint sets Theta_0 and Theta_1. Let H_0 denote the hypothesis that heta in Theta_0, and let H_1 denote the hypothesis that heta in Theta_1.The binary test of hypotheses is performed using a test function phi(x). :phi(x) = egin{cases} 1 & ext{if } x in R \0 & ext{if } x in Aend{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 A. A cup R is a disjoint covering of the measurement space.

Formal definition

A test function phi(x) is UMP of size alpha if for any other test function phi'(x) we have::sup_{ hetainTheta_0}; E_ hetaphi'(X)=alpha'leqalpha=sup_{ hetainTheta_0}; E_ hetaphi(X),: E_ hetaphi'(X)=1-eta'leq 1-eta=E_ hetaphi(X) quad forall heta in Theta_1

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 parameter "&theta;", and define the likelihood ratio l(x) = f_{ heta_1}(x) / f_{ heta_0}(x).If l(x) is monotone non-decreasing for any pair heta_1 geq heta_0 (meaning that the greater x is, the more likely H_1 is), then the threshold test::phi(x) = egin{cases} 1 & ext{if } x > x_0 \0 & ext{if } x < x_0end{cases}:E_{ heta_0}phi(X)=alphais the UMP test of size "α" for testing H_0: heta leq heta_0 ext{ vs. } H_1: heta > heta_0

Note that exactly the same test is also UMP for testing H_0: heta = heta_0 ext{ vs. } H_1: heta > heta_0

Important case: The exponential family

Although the Karlin-Rubin 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 f_ heta(x) = c( heta)h(x)exp(pi( heta)T(x)) has a monotone non-decreasing likelihood ratio in the sufficient statistic "T"("x"), provided that pi( heta) is non-decreasing.


Let X=(X_0 , X_1 ,dots , X_{M-1}) denote i.i.d. normally distributed N-dimensional random vectors with mean heta m and covariance matrix R. We then have :f_ heta (X) = (2 pi)^{-M N / 2} |R|^{-M / 2} exp left{-frac{1}{2} sum_{n=0}^{M-1}(X_n - heta m)^T R^{-1}(X_n - heta m) ight} = : = (2 pi)^{-M N / 2} |R|^{-M / 2} exp left{-frac{1}{2} sum_{n=0}^{M-1}( heta^2 m^T R^{-1} m) ight} cdot exp left{-frac{1}{2} sum_{n=0}^{M-1}X_n^T R^{-1} X_n ight} cdot exp left{ heta m^T R^{-1} sum_{n=0}^{M-1}X_n ight}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:phi(T) = egin{cases} 1 & ext{if } T > t_0 \0 & ext{if } T < t_0end{cases}:E_{ heta_0} phi (T) = alpha

is the UMP test of size alpha for testing H_0: heta leq heta_0 vs. H_1: heta > heta_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). Why is it so?

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 heta_1 where heta_1 > heta_0) is different than the most powerful test of the same size for a different value of the parameter (e.g. for heta_2 where heta_2 < heta_0). As a result, no test is Uniformly most powerful.


* L. L. Scharf, "Statistical Signal Processing", Addison-Wesley, 1991, section 4.7.

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