An empirical adjustment to the likelihood ratio statistic

Consider a model parameterised by a scalar parameter of interest psi and a nuisance parameter lambda. Inference about psi may be based on the signed s quare root of the likelihood ratio statistic, R. The statistic R is asympto tically distributed according to a standard normal distribution, with error O(n(-1/2)). To reduce the error of this normal approximation, several modi fications to R have been proposed such as Barndorff-Nielsen's modified dire cted likelihood statistic, R*. In this paper, an approximation to R* is pro posed that can be calculated numerically for a wide range of models. This a pproximation is shown to agree with R* with error of order O-p(n(-1)). The results are illustrated on several examples.