Reference:
Theoretical Statistics by Robert W. Keener
Probability Theory by Eiko
Foundations of Modern Probability by Olav Kallenberg
A test \(\varphi^*\) with level \(\alpha_{\varphi^*} = \alpha\) is called uniformly most powerful if for any \(\theta\in \Omega_1\), we have
\[\mu_\theta(\varphi^*)\ge \sup_{\alpha_\varphi\le \alpha} \mu_\theta(\varphi) \quad \forall \theta\in \Omega_1.\]
This means we are not just optimizing for one \(\theta_1\) nor for the power function, we are optimizing for all \(\theta_1\in \Omega_1\). It might not exist.
A family of densities \(p_\theta(x)\), \(\theta\in \Omega\subset \mathbb{R}\) has monotone increasing likelihood ratios if there exists a statistic \(T=T(x)\) such that whenever \(\theta_1<\theta_2\), the likelihood ratio \(p_{\theta_2}(x)/p_{\theta_1}(x)\) is a non-decreasing function of \(T(x)\).
For the decreasing case it suffices to replace \(T\) by \(-T\).
Example. For densities of the form of exponential families (of single parameter)
\[\mu_\theta \sim \exp\left(\eta(\theta)T(x) - B(\theta)\right)h(x) \,\mathrm{d}{x},\]
if \(\eta(\theta)\) is non-decreasing / increasing, the likelihood ratio
\[ L(x) = \frac{\mathrm{d} \mu_{\theta_2}}{\mathrm{d} \mu_{\theta_1}} = \exp\left((\eta(\theta_2)-\eta(\theta_1))T(x) - B(\theta_2)+B(\theta_1)\right) \]
is a non-decreasing / increasing function of \(T(x)\). This includes the example of the exponential distribution, for which \(\eta(\theta) = \theta\) is increasing and \(T(x) = -x\).
For this concept of monotone likelihood, we assume the parameter space \(\theta\in \Omega\subset \mathbb{R}\) is one-dimensional.
Suppose the family of density \(\mathbb{P}(X|\theta)\) has monotone likelihood ratio (with respect to a test statistic \(T(x)\)). Then
The monotone \(T\) test
\[\varphi^*(x) = \begin{cases} 1, & T>c \\ \gamma, & T=c \\ 0, & T<c \end{cases} \]
is a uniformly most powerful test for the hypothesis \(H_0: \theta\le \theta_0\) and \(H_1: \theta>\theta_0\), with level
\[\alpha = \mu_{\theta_0}\varphi^*.\]
\(c\) and \(\gamma\) can be adjusted to achieve any desired level \(\alpha\).
The power function \(\beta(\theta)=\mu_\theta(\varphi^*)\) for this test is strictly increasing when it is in \((0,1)\), and non-decreasing overall.