  By the modified directed likelihood, higher order accurate confidence limits
for a scalar parameter are obtained from the likelihood. They are conveniently
described in terms of a confidence distribution, that is a sample dependent
distribution function on the parameter space. In this paper we explore a
different route to accurate confidence limits via tail-symmetric confidence
curves, that is curves that describe equal tailed intervals at any level.
Instead of modifying the directed likelihood, we consider inversion of the
log-likelihood ratio when evaluated at the median of the maximum likelihood
estimator. This is shown to provide equal tailed intervals, and thus an exact
confidence distribution, to the third-order of approximation in regular
one-dimensional models. Median bias correction also provides an alternative
approximation to the modified directed likelihood which holds up to the second
order in exponential families.
