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 T _ψ or the conditional likelihood given a function S _ψ. The properties and construction of marginal and conditional likelihood functions based on parameter-dependent functions are studied. In particular, the case in which T_ψ and S_ψ 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.