Integrated likelihoods for functions of a parameter

Non‐Bayesian inference regarding a parameter of interest in the presence of a nuisance parameter may be based on the integrated likelihood function, in which the nuisance parameter is eliminated by averaging the likelihood function with respect to a weight function for the nuisance parameter. Recent research has shown that integrated likelihood methods work particularly well when the model is reparameterized in terms of a nuisance parameter chosen to be “unrelated” to the parameter of interest and the corresponding weight function for this nuisance parameter is chosen so that it does not depend on the parameter of interest. One choice for such a nuisance parameter is the zero score expectation (ZSE) parameter. The purpose of this note is to extend the definition of the ZSE parameter to the case in which the model has a parameter vector θ, with the parameter of interest of the model taken to be a function of θ; that is, the definition of the ZSE parameter is extended to the case in which there is not an explicit nuisance parameter for the model. The resulting integrated likelihood function has the same desirable properties as integrated likelihoods based on the ZSE parameter in models with an explicit nuisance parameter.