Karlin–Rubin Theorem

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 $f_(x)$, 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).
:$\varphi(x) = \begin 1 & \text x \in R \\ 0 & \text x \in R^c \end$
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_\; \operatorname \theta\alpha'\leq\alpha=\sup_\; \operatorname \theta,
we have
: \forall \theta \in \Theta_1, \quad \operatorname \theta 1 - \beta'(\theta) \leq 1 - \beta(\theta) =\operatorname \theta

The Karlin–Rubin theorem

The Karlin–Rubin theorem can be regarded as an extension of the Neyman–Pearson lemma for composite hypotheses.Casella, G.; Berger, R.L. (2008), ''Statistical Inference'', Brooks/Cole. (Theorem 8.3.17) Consider a scalar measurement having a probability density function parameterized by a scalar parameter ''θ'', and define the likelihood ratio $l(x) = f_(x) / f_(x)$.
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:
:$\varphi(x) = \begin 1 & \text x > x_0 \\ 0 & \text x < x_0 \end$
:where $x_0$ is chosen such that $\operatorname_\varphi(X)=\alpha$

is the UMP test of size ''α'' for testing $H_0: \theta \leq \theta_0 \text H_1: \theta > \theta_0 .$

Note that exactly the same test is also UMP for testing $H_0: \theta = \theta_0 \text H_1: \theta > \theta_0 .$

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
:$f_\theta(x) = g(\theta) h(x) \exp(\eta(\theta) T(x))$
has a monotone non-decreasing likelihood ratio in the sufficient statistic $T(x)$, provided that $\eta(\theta)$ is non-decreasing.

Example

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

:\begin
f_\theta (X) = & (2 \pi)^ , R, ^ \exp \left\ \\ pt= & (2 \pi)^ , R, ^ \exp \left\ \\ pt& \exp \left\ \exp \left\
\end

which is exactly in the form of the exponential family shown in the previous section, with the sufficient statistic being

: $T(X) = m^T R^ \sum_^X_n.$

Thus, we conclude that the test
:$\varphi(T) = \begin 1 & T > t_0 \\ 0 & T < t_0 \end \qquad \operatorname_ \varphi (T) = \alpha$

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.

References

Further reading

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

{{DEFAULTSORT:Uniformly Most Powerful Test
Statistical hypothesis testing