Stable and invariant adjusted profile likelihood and directed likelihood for curved exponential models

For curved exponential models, the modified profile likelihoodand the modified directed likelihood are approximated, witherror 0(n^-1) under ordinary repeated sampling, by explicitlydefined and invariant quantities which do not require specificationof an ancillary and which are stable in the sense of havingthe appropriate conditionality property with respect to anyreasonable ancillary.