A NOTE ON THE RELATION BETWEEN MODIFIED PROFILE LIKELIHOOD AND THE COX-REID ADJUSTED PROFILE LIKELIHOOD

An adjustment to the profile likelihood proposed by Cox & Reid (1987) when the parameters are orthogonal is shown to agree with modified pro file likelihood in a number of instances in which the parameters are n ot orthogonal. Several examples of the phenomenon are given. In the ca se of composite transformation models with invariant parameter as the parameter of interest, it is shown that the Cox-Reid adjustment agrees with modified profile likelihood in a large-deviation sense if Lebesg ue measure is right-invariant. Necessary and sufficient conditions for equivalence in the moderate-deviation sense of modified profile likel ihood and the Cox-Reid adjusted profile likelihood are derived.