On the small sample distributions of the empirical loglikelihood ratio

The author shows that the empirical loglikelihood ratio is also pivotal for the multivariate normal mean and derives E distributions, a family of distributions equivalent to Hotelling’s T 2 in the context of empirical likelihood. For empirical likelihood ratio confidence regions based on estimating equations, the author discusses bounds on their coverage levels derived from the atoms of the E distributions. The author also introduces E calibration which calibrates a general empirical loglikelihood ratio using the E distributions. Discussions on the atoms and bounds provide further understanding on the undercoverage problem of the asymptotic Chi-square calibrated empirical likelihood ratio confidence regions. The E calibration respects bounds on coverage levels and is a simple, effective method for correcting the undercoverage problem.