likelihood inference with no MLE

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“In a regular full discrete exponential family, the MLE for the canonical parameter does not exist when the observed value of the canonical statistic lies on the boundary of its convex support.”

Daniel Eck and Charlie Geyer just published an interesting and intriguing paper on running efficient inference for discrete exponential families when the MLE does not exist.  As for instance in the case of a complete separation between 0’s and 1’s in a logistic regression model. Or more generally, when the estimated Fisher information matrix is singular. Not mentioning the Bayesian version, which remains a form of likelihood inference. The construction is based on a MLE that exists on an extended model, a notion which I had not heard previously. This model is defined as a limit of likelihood values

\lim_{n\to\infty} \ell(\theta_n|x) = \sup_\theta \ell(\theta|x) := h(x)

called the MLE distribution. Which remains a mystery to me, to some extent. Especially when this distribution is completely degenerate. Examples provided within the paper alas do not help, as they mostly serve as illustration for the associated rcdd R package. Intriguing, indeed!

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