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

Abstract

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 Θ 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 Θ has a discrete, but not necessarily lattice, distribution. If Θ has a lattice distribution, then p * provides a valid approximation to the density of Θ with respect to counting measure. If the distribution of Θ is non‐lattice, then p * is still a valid approximation to the conditional density of Θ however, the dominating measure is no longer counting measure.