FREQUENTIST AND BAYESIAN BARTLETT CORRECTION OF TEST STATISTICS BASEDON ADJUSTED PROFILE LIKELIHOODS

For inference about a multidimensional parameter psi in the presence o f nuisance parameters, the usual log-profile-likelihood function M(psi ) is often replaced by an objective function M(psi)BAR = M(psi) + B(ps i), where B(psi) is O(p)(1). It is shown in this paper that under mild conditions on the adjustment function B(psi) both frequentist and Bay esian Bartlett correction are possible for the adjusted likelihood rat io statistic arising from M(psi)BAR. General formulae for the Bartlett adjustment factors are developed, and these formulae are evaluated fo r some specific adjustment functions B(psi). Furthermore, conditions u nder which the frequentist and Bayesian adjustment factors agree to or der O(p)(n-3/2) are considered. In particular, prior probability densi ty functions for which the highest posterior density region of posteri or probability content 1 - alpha has frequentist coverage level 1 - al pha + 0 (n-2 ) are characterized, and similar characterizations are gi ven for likelihood-based regions.