Adjusted Versions of Profile Likelihood and Directed Likelihood, and Extended Likelihood

SUMMARY Several adjusted and invariant versions of profile likelihood and of directed likelihood (short for signed log-likelihood ratio) are discussed. These adjusted versions, which fall into two groups, are closely related to modified profile likelihood and modified directed likelihood. The first group has the advantage of not requiring an explicit specification of an ancillary, but partly at a cost of a loss of asymptotic accuracy in comparison with the modified versions. The second group consists of some asymptotically sharpened versions of modified profile likelihood. Furthermore, a concept of extended likelihood is proposed, which associates a likelihood-like function to certain combinants. At the end, the relationship between adjusted profile likelihoods on the one hand and adjusted directed likelihoods on the other is discussed briefly, and in this context a close connection is established between modified profile likelihood and the extended likelihood determined by the modified directed likelihood.