Modified signed log likelihood ratio

Abstract

SUMMARY The signed log likelihood ratio r for a one-dimensional interest parameter can be modified as r * = r + r −1 log (u/r) so that r * is asymptotically standard normally distributed with error of order O(n −/32; ), u being interpretable as a test statistic. A new, more direct derivation of this result is given, which leads to a more explicit expression for u and provides a better handle on the error term. The relation of the tail area approximation determined by r * to a generalization of the Lugannani-Rice approximation is discussed, and the applicability of r * to the analysis of residual variation is indicated.