On standardizing the signed root log likelihood ratio statistic

A simple connection between the Bartlett adjustment factor of the log likelihood ratio statistic and the normalizing constant of the p∗ formula–an approximate conditional density for the maximum likelihood estimate given an exact or an approximate ancillary statistic–was established in Barndorff-Nielsen and Cox (1984). In this paper, the explicit form of the normalizing constant of the p∗ formula for the scalar parameter model is derived. By change of variables, the mean and variance of the signed root log likelihood ratio statistic are obtained explicitly, and, hence, tail probabilities can be calculated from the standardized signed root log likelihood ratio statistic. Examples are used to illustrate the implementation and accuracy of the proposed method.