Approximate Interval Probabilities

For models of parametric dimension 1 two parameterization invariant approximations to the conditional distribution function of the maximum likelihood estimator are considered. The first is derived by partial integration of a formula for the conditional density given an ancillary statistic, a technique which in addition yields a useful expression for the error term. The second approximation is based on an adjusted version of the signed log-likelihood ratio statistic. A third approximation is a modification of the first that avoids the specification of an ancillary