Saddlepoint approximation for the distribution of the modified signed root of likelihood ratio statistics near the mean

Barndorff-Nielsen introduced a variable U in the adjusted saddlepoint approximation (R)+(R)(R-1-U-1) to reduce the order of error in the standard normal approximation to the distribution of the log-likelihood ratio statistic R. DiCiccio and Martin (1993) proposed a modification, which avoids the practical difficulties of finding an exact or approximate ancillary A in calculating Barndorff-Nielsen's variable U. We repair the numerical difficulties in applying their modification, when the ordinate at which the approximation is evaluated is near the conditional mean of the approximated distribution. Some special cases and examples are discussed.