  The use of standard statistical methods, such as maximum likelihood, is often
justified based on their asymptotic properties. For suitably regular models,
this theory is standard but, when the model is non-regular, e.g., the support
depends on the parameter, these asymptotic properties may be difficult to
assess. Recently, an inferential model (IM) framework has been developed that
provides valid prior-free probabilistic inference without the need for
asymptotic justification. In this paper, we construct an IM for a class of
highly non-regular models with parameter-dependent support. This construction
requires conditioning, which is facilitated through the solution of a
particular differential equation. We prove that the plausibility intervals
derived from this IM are exact confidence intervals, and we demonstrate their
efficiency in a simulation study.
