Exact prior-free probabilistic inference in a class of non-regular models

TL;DR

This paper develops an exact, prior-free probabilistic inference method for highly non-regular models with parameter-dependent support, using an inferential model framework that ensures valid confidence intervals without asymptotic assumptions.

Contribution

It introduces a novel IM construction for non-regular models requiring differential equation solutions, providing exact confidence intervals and demonstrating efficiency.

Findings

Constructed an IM for non-regular models with parameter-dependent support.

Proved that the plausibility intervals are exact confidence intervals.

Demonstrated the method's efficiency through simulation studies.

Abstract

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.