High-order asymptotics for tail symmetry of confidence curves

Confidence curves are stochastic functions taking values into the unit interval whose level sets provide confidence regions for the unknown parameter. Inversion of the likelihood ratio test leads, via a probability integral transformation, to confidence curves which point at the maximum likelihood estimate. For a one-dimensional parameter, the allied confidence intervals generally do not have equal-tail probabilities. We consider a correction to the log-likelihood ratio which leads to confidence curves that are asymptotically tail-symmetric to the third order of approximation. This happens provided that the maximum likelihood estimator is distributed according to Efron’s normal transformation family.