Likelihood functions based on parameter-dependent functions

Consider likelihood inference about a scalar function ψ of a parameter θ. Two methods of constructing a likelihood function for ψ are conditioning and marginalizing. If, in the model with ψ held fixed, T is ancillary, then a marginal likelihood may be based on the distribution of T, which depends only on ψ; alternatively, if a statistic S is sufficient when ψ is fixed, then a conditional likelihood function may be based on the conditional distribution of the data given S. The statistics T and S are generally required to be the same for each value of ψ. In this paper, we consider the case in which either T or S is allowed to depend on ψ. Hence, we might consider the marginal likelihood function based on a function Tψ or the conditional likelihood given a function Sψ. The properties and construction of marginal and conditional likelihood functions based on parameter-dependent functions are studied. In particular, the case in which Tψ and Sψ may be taken to be functions of the maximum likelihood estimators is considered and approximations to the resulting likelihood functions are presented. The results are illustrated on several examples.