Define likelihoods for estimation in non-dominated models

Develop a general framework for defining a likelihood function that supports estimation procedures such as maximum likelihood estimation in statistical model families that lack a common dominating measure, for example when comparing or combining discrete and continuous distributions, where no hypothesis-based construction (such as an effective null) is available.

Background

The paper critiques a universalist commitment to Likelihoodism by highlighting settings where standard likelihood-based tools are ill-defined. When a model family contains distributions that are not jointly dominated by a single measure (e.g., comparing a continuous distribution against a discrete distribution), the usual construction of the likelihood function breaks down.

While recent work proposes an effective-null construction to enable hypothesis testing in such settings, the authors emphasize that estimation problems (e.g., maximum likelihood estimation) remain unresolved because there are no hypotheses to anchor the construction. This leaves an explicit gap: a general approach to define likelihoods for estimation in non-dominated models.