Adjustments to profile likelihood

Conditional and marginal likelihoods constructed from parameter-dependent functions of the data have an additional parameter dependence that changes with the choice of supporting metric for the corresponding densities. We consider constructing marginal and conditional likelihoods by using densities expressed in terms of an instrinsic choice of support measure, based on either Euclidean metric or the constant information metric. Barndorff-Nielsen's (1983) approximation to the distribution of the maximum likelihood estimate is used to approximate these conditional and marginal likelihood of Barndorff-Nielsen (1983) and the approximate conditional likelihood of Cox & Reid (1987). The normal circle problem is examined in some detail.