A note on directed adjusted profile likelihoods

Several adjustments to the profile likelihood have been proposed in recent years, to take into proper account the effects of fitting nuisance parameters. In some cases, adjusted profile likelihoods are higher-order approximations of suitable conditional or marginal target likelihoods. However, the adjustments seem to provide accurate inference also when an exact marginal or conditional target likelihood is not available. Here, we consider adjusted profile likelihoods as approximations of a suitable general target likelihood. This is the likelihood for the parameter of interest with a known orthogonal nuisance parameter. Attention is focused on a scalar parameter of interest. Some new results are obtained concerning the null and non-null distributions of the directed likelihood calculated from an adjusted profile likelihood. In particular, we show that, while these distributions match the corresponding null and non-null distributions of the directed likelihood computed from the target likelihood up to order O(n^( 121/2)) included, the agreement does not in general carry over to terms of order O(n^( 121)), even if the information bias is of order O(n^( 121))