Testing a Null Hypothesis Nested by the Alternative Hypothesis

Consider a parameter of interest $$\beta \in \mathcal{B}$$, and two hypotheses $$H_0:\;\beta\in\mathcal{B_0}\quad versus \quad H_1:\;\beta\in\mathcal{B}_1$$ where $$\mathcal{B}_0\cup\mathcal{B_1}=\mathcal{B}$$ and $$\mathcal{B}_0\subset\mathcal{B_1}$$.

That is, the null hypothesis is a subset of the alternative hypothesis.

To be specific, I am dealing with hypotheses $$H_0:\;\beta\geq0\quad versus \quad H_1:\;\beta\geq c$$ where $$c$$ is an unknown negative constant (i.e. $$c<0$$ and $$c$$ is unknown).

Is there any method or approach related to this situation?

(this post is closely related to my past posting Testing a special inequality hypothesis)

• Conventionally, the class of distributions $\mathcal P$ is classified into two mutually exclusive subclasses $\mathcal H_0\uplus \mathcal H_1= \mathcal P.$ That is the basis of hypothesis testing. The current situation is pretty unorthodox and it seems evident there would be cases where both would be true. Sep 22, 2022 at 4:45
• @User1865345 Not only mutually exclusive, but complementary. Sep 22, 2022 at 5:39

If you are interested in the probability of $$H_0$$ compared to the probability of $$H_1$$, you might want to approximate the density and then integrate it over both $$H_0$$ and $$H_1$$.
• @Glen_b the simple-vs-simple hypotheses consider population families with just two members, so in this sense, all "possibilities" (i.e. all populations in the considered family) are contained in either $H_0$ or $H_1$. See also the first paragraph of chapter 6 in Jun Shao's "Mathematical Statistics". Sep 22, 2022 at 8:00
• @Glen_b I agree that in general we only need disjointness. E.g. Young and Smith (2005) write on page 65 "Throughout we have a parameter space $\Theta$, and consider hypotheses of the form $H_0:\theta \in \Theta_0$ vs. $H_1:\theta \in \Theta_1$, where $\Theta_0$ and $\Theta_1$ are two disjoint subsets of $\Theta$, possibly, but not necessarily, satisfying $\Theta_0 \cup\Theta_1=\Theta$." Sep 22, 2022 at 10:22
• @statmerkur But what is then $\Theta \setminus (\Theta_0 \cup \Theta_1)$ used for? Why not setting $\Theta = \Theta_0 \cup \Theta_1$ ? Sep 22, 2022 at 14:13