© 1983 by Biometrika Trust
On a formula for the distribution of the maximum likelihood estimator
Department of Theoretical Statistics, Aarhus University Aarhus, Denmark
A simple formula for the conditional distribution of the maximum likelihood estimator given a maximal ancillary statistic is discussed and exemplified. The formula is generally accurate to order O(n–
l) or even O(n–3/2), and for many important models it is, in fact, exact. After some preliminary discussion of the formula and of certain relevant aspects of likelihood, the formula is used to motivate the definition of a modified profile likelihood whose inferential properties are illustrated. The question of when the distribution formula is exact is considered, and in this connexion several new examples of exactness, including a bivariate generalization of the inverse Gaussian distribution, are adduced. The formula is shown also to be exact for arbitrary transformation models. To prove this it has been necessary to extend the basic theory of transformation models to cover the cases where the group action is not free. This extension, which appears of interest in itself, also allows of a generalization of a useful formula for the marginal likelihood for the index parameter of a composite transformation model.
Key Words: Ancillarity • Conditionality • Elimination of nuisance parameters • Hyperboloid model • Information • Invariant measure • Inverse Gaussian model • Marginal likelihood • Maximal invariant • Modified profile likelihood • Quasilikelihood • Saddlepoint approximation • Stable distribution • Transformation model
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