The likelihood ratio approximation to the conditional distribution of the maximum likelihood estimator in the discrete case

The likelihood ratio approximation, also called Barndorff-Nielsen's approximation and often denoted by p*, provides a highly accurate approximation to the conditional density of a maximum likelihood estimator θ̂ given an ancillary statistic. In this paper, the properties of p* are considered for the case in which the underlying random variables have a lattice distribution and θ̂ has a discrete, but not necessarily lattice, distribution. If θ̂ has a lattice distribution, then p* provides a valid approximation to the density of θ̂ with respect to counting measure. If the distribution of θ̂ is non-lattice, then p* is still a valid approximation to the conditional density of θ̂; however, the dominating measure is no longer counting measure.