Uniformly most powerful test

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 & ~~ \text{ if } ~~ x \in R \ 0 & ~~ \text{ if } ~~ x \in R^\mathsf{;! 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^\mathsf{;! 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, \quad \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 (named for Samuel Karlin and Herman Rubin) 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 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 & \text{if } x x_0 \ 0 & \text{if } x \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 \text{ vs. } H_1: \theta \theta_0 .

Note that exactly the same test is also UMP for testing H_0: \theta = \theta_0 \text{ 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(\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_{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{-\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{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_{n=0}^{M-1}X_n.

Thus, we conclude that the test :\varphi(T) = \begin{cases} 1 & T t_0 \ 0 & T

is the UMP test of size \alpha for testing H_0: \theta \leqslant \theta_0 vs. H_1: \theta \theta_0

Further discussion

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 ). As a result, no test is uniformly most powerful in these situations.

References

References

1. Casella, G.; Berger, R.L. (2008), ''Statistical Inference'', Brooks/Cole. {{ISBN. 0-495-39187-5 (Theorem 8.3.17)