An empirical adjustment to the likelihood ratio statistic

Consider a model parameterised by a scalar parameter of interest ψ and a nuisance parameter λ Inference about ψ may be based on the signed square root of the likelihood ratio statistic, R. The statistic R is asymptotically distributed according to a standard normal distribution, with error O(n^-1/2). To reduce the error of this normal approximation, several modifications to R have been proposed such as Barndorff-Nielsen's modified directed likelihood statistic, R*. In this paper, an approximation to R* is proposed that can be calculated numerically for a wide range of models. This approximation is shown to agree with R* with error of order O_p(n^-1). The results are illustrated on several examples.