Likelihood-ratio tests with boundary randomization

Let $\mathsf{E}=((\mathcal{X},\mathcal{A}),\{0,1\},(P_0,P_1))$ be the two-law statistical experiment (Statistical experiments). Assume $P_0,P_1$ are both absolutely continuous with respect to a supplied sigma-finite measure $\mu$ on $(\mathcal{X},\mathcal{A})$ (Dominated statistical experiments) and specify finite-valued, nonnegative measurable RN density representatives $p_0=d{P}_0\mathrm{/}d\mu$ and $p_1=d{P}_1\mathrm{/}d\mu$ (Likelihood densities). Under the Axiom of Choice (The Axiom of Choice), Likelihood densities supplies such representatives for arbitrary dominated laws. If the representatives are given as part of the setup, the construction below requires no choice.

Define the common-support likelihood ratio $R(x)=\{\begin{array}{ll}{p}_1(x)\mathrm{/}{p}_0(x),& p_0(x)>0,\\ 0,& p_0(x)=0.\end{array}$ The result The common-support likelihood ratio is measurable shows that $R$ is finite-valued, nonnegative, and measurable. The set $\{p_0=0<p_1\}$ is kept separate from the ratio region.

For $\alpha =0$, define the rejection function ${\phi }_0^{\ast }(x)={\mathbf{1}}_{\{p_0=0<p_1\}}(x)\mathrm{.}$ For $0<\alpha \le 1$, choose $c\ge 0$ and $\gamma \in [0,1]$ satisfying $P_0(R>c)+\gamma P_0(R=c)=\alpha ,$ whose existence follows from A likelihood-ratio threshold with boundary randomization exists applied to $Q=P_0$ and $T=R$. Define

This specifies the candidate likelihood-ratio rejection function.

The common dominating measure and the specified density representatives are part of the setup. The general density-selection branch uses the Axiom of Choice through Likelihood densities; with explicit representatives, the cutoff existence result and displayed construction are choice-free.