This paper considers the problem of constructing approximate confidence intervals for functional parameters in the nonparametric case. The approach based on transformation theory is applied to improve standard confidence intervals. The accelerated bias-corrected percentile interval introduced by Efron relies on the existence of a normalizing transformation with bias and skewness corrections, although calculation does not require explicit knowledge of its functional form. We formally construct such a transformation and estimate bias and skewness correction factors for nonparametric situations. The resulting interval is shown to be second-order accurate. To this end Edgeworth expansions for the distributions of transformed statistics are derived, using the von Mises expansion.
"Normalizing Transformatins and Bootstrap Confidence Intervals." Ann. Statist. 19 (4) 2209 - 2225, December, 1991. https://doi.org/10.1214/aos/1176348393