On expected volumes of multidimensional confidence sets associated with the usual and adjusted likelihoods

We consider a general multiparameter set-up, where both the interest and th e nuisance parameters are possibly vector valued. We derive an explicit hig her order asymptotic formula to compare the expected volumes of confidence sets given by likelihood ratio statistics arising from the usual profile li kelihood and various adjustments thereof. Our general framework also allows us to include highest posterior density regions, with approximate frequent ist validity, in the study. The fact that our interest parameter Is possibl y vector valued complicates the derivation and warrants the development of special tools and techniques.