Uniformly most powerful test explained

In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power

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

X denote a random vector (corresponding to the measurements), taken from a parametrized family of probability density functions or probability 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 & if x \in R \\ 0 & 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,} \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

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 & if x > x _{0} \\ 0 & 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 size α for testing

H _{0:} \theta \leq \theta _{0} vs. H _{1:} \theta > \theta _{0} .

Note that exactly the same test is also UMP for testing

H _{0:} \theta = \theta _{0} 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

f _{\theta(x)} = g(\theta) h(x) \exp( η (\theta) T(x)) has a monotone non-decreasing likelihood ratio in the sufficient statistic T(x) , provided that η (\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{align} f _{\theta} (X) = {} & ( 2 \pi) ^{-MN/2} |R| ^{-M/2} \exp \left\{ - (1)/(2) \sum ^{M -1} _{n =0} (X _{n} - \theta m) ^{T} R ^{-1} (X _{n} - \theta m) \right\} \\[ 4 pt] = {} & ( 2 \pi) ^{-MN/2} |R| ^{-M/2} \exp \left\{ - (1)/(2) \sum ^{M -1} _{n =0} \left (\theta ^{2} m ^{T} R ^{-1} m \right ) \right\} \\[ 4 pt] & \exp \left\{ - (1)/(2) \sum ^{M -1} _{n =0} X ^{T} _{n} R ^{-1} X _{n} \right\} \exp \left\{\theta m ^{T} R ^{-1} \sum ^{M -1} _{n =0} X _{n} \right\} \end{align}

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 ^{M -1} _{n =0} X _{n.}

Thus, we conclude that the test

\varphi(T) = \begin{cases} 1 & T > t _{0} \ 0 & T < t _{0} \end{cases} \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 is uniformly most powerful in these situations.

Further reading

• Book: Ferguson. T. S.. Thomas S. Ferguson . Mathematical Statistics: A decision theoretic approach. Academic Press. New York. Sec. 5.2: Uniformly most powerful tests. 1967.
• Book: Mood. A. M.. Graybill. F. A.. Boes. D. C.. Introduction to the theory of statistics. 3rd. McGraw-Hill. New York. Sec. IX.3.2: Uniformly most powerful tests. 1974.
• L. L. Scharf, Statistical Signal Processing, Addison-Wesley, 1991, section 4.7.

Notes and References

1. Casella, G.; Berger, R.L. (2008), Statistical Inference, Brooks/Cole. (Theorem 8.3.17)