Modified signed log likelihood ratio

Abstract

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.