Bartlett Adjustments to the Likelihood Ratio Statistic and the Distribution of the Maximum Likelihood Estimator

For rather general parametric models, a simple connection is established between the Bartlett adjustment factor of the log-likelihood ratio statistic and the normalizing constant c of the formula c I I 1?2L for the conditional distribution of a maximum likelihood estimator as applied to the full model and the model of the hypothesis tested. This leads to a relatively simple demonstration that division of the likelihood ratio statistic by a suitable constant or estimated factor improves the chi-squared approximation to its distribution. Various expressions for these quantities are discussed. In particular, for the case of a one-dimensional parameter an approximation to the constants involved is derived, which does not require integration over the sample space.