Consider a model with parameter \(\theta \) and likelihood function \(L(\theta )\) . Suppose that \(\theta \) can be written \(\theta = (\psi, \lambda )\) , where \(\psi \) is a real-valued parameter-of-interest and \(\lambda \) is a nuisance parameter and that our goal is likelihood-based inference regarding \(\psi \) . In some cases, it may be beneficial to reparameterize the model, keeping the parameter-of-interest unchanged but modifying the nuisance parameter of the model. The purpose of this paper is to present some results regarding a specific nuisance parameter, known as the zero-score expectation (ZSE) parameter, that has been shown to be useful in integrated likelihood inference. These results include a more straightforward motivation for the ZSE parameter, an approximation useful for calculating the corresponding likelihood function, and a new interpretation of an integrated likelihood constructed using the ZSE parameter.
