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)

By definition (see e.g. Jun Shao "Mathematical Statistics", 2nd edition, chapter 6, first paragraph) of statistical hypothesis tests, the null and the alternative hypothesis must be disjoint. Moreover, both hypotheses together must contain all possibilities.
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 $H_1$.
