Finite type-II testing function with explicit orientation

Let $P,Q$ be probability vectors on a finite nonempty alphabet $\mathcal{X}$ (Finite probability vectors). They define finite probability spaces with outcome weights $p_x$ and $q_x$, respectively (Finite probability spaces, outcome weights, events, and event probabilities). For $h:\mathcal{X}\to \mathbb{R}$, write $E_{P}h:={\sum }_x{p}_{x}h(x)$ and $E_{Q}h:={\sum }_x{q}_{x}h(x)$; these are the finite expectations (Expectation of a real random variable on a finite probability space). Test the simple null $H_0:P$ against the simple alternative $H_1:Q$. A randomized test is a function $\varphi :\mathcal{X}\to [0,1]$, where $\varphi (x)$ is the probability of rejecting $H_0$ after observing $x$; thus $1-\varphi (x)$ is the probability of accepting $H_0$ (Randomized tests). Its type-I error is ${\sum }_x{p}_x\varphi (x)$, and its type-II error is ${\sum }_x{q}_x(1-\varphi (x))$ (Test size and power).

For $ϵ\in [0,1]$, define the oriented type-II testing function ${\beta }_{ϵ}(P,Q):={\inf }_{\begin{array}{c}\varphi :\mathcal{X}\to [0,1]\\ {\sum }_x{p}_x\varphi (x)\le ϵ\end{array}}{\sum }_x{q}_x(1-\varphi (x))\mathrm{.}$ Here the first argument is the null law and the second is the alternative law; $ϵ$ bounds rejection under the first law, while the objective is acceptance of the null under the second. The feasible set is nonempty because $\varphi =0$ is allowed, and every objective value lies in $[0,1]$. Thus the set of objective values is nonempty and bounded below in $\mathbb{R}$, so it has a real infimum: The real numbers identifies $\mathbb{R}$ with the Cauchy-sequence reals, The reals form a totally ordered field gives their ordered-field structure, and The Cauchy-sequence reals have the least-upper-bound property gives their least-upper-bound property. Complete ordered field (least-upper-bound property) gives the equivalent infimum property. Since the set also lies in $[0,1]$, that infimum lies in $[0,1]$. In the Neyman–Pearson-region convention that constrains null acceptance below by $\gamma$, this is the lower boundary value at $\gamma =1-ϵ$.

The symbol ${\beta }_{ϵ}(P,Q)$ here is the infimum type-II error. It is not the power function $E_{\theta }[\varphi ]$ from Test size and power; under $Q$, it minimizes one minus that power.