Frequentist and Bayesian Bartlett Correction of Test Statistics Based on Adjusted Profile Likelihoods

SUMMARY For inference about a multidimensional parameter ψ in the presence of nuisance parameters, the usual log-profile-likelihood function M(ψ) is often replaced by an objective function M̄(ψ) = M(ψ) + B(ψ), where B(ψ) is Op(1). It is shown in this paper that under mild conditions on the adjustment function B(ψ) both frequentist and Bayesian Bartlett correction are possible for the adjusted likelihood ratio statistic arising from M̄(ψ). General formulae for the Bartlett adjustment factors are developed, and these formulae are evaluated for some specific adjustment functions B(ψ). Furthermore, conditions under which the frequentist and Bayesian adjustment factors agree to order Op(n –3/2) are considered. In particular, prior probability density functions for which the highest posterior density region of posterior probability content 1 – α has frequentist coverage level 1 – α + O(n –2) are characterized, and similar characterizations are given for likelihood-based regions.