Miscellanea. The directed modified profile likelihood in models with many nuisance parameters

The modified profile likelihood is a higher-order approximation to a conditional or a marginal likelihood, when either exists. The directed likelihood based on the modified profile likelihood is standard normal only to first order. In practice, however, it usually achieves most of the distributional improvement obtained by higher-order methods. This note elucidates this point with particular attention to models with many nuisance parameters.