Likelihood Inference in Exponential Families and Directions of Recession

TL;DR

This paper addresses the challenge of likelihood inference in exponential families when the MLE does not exist by proposing an algorithm to find the MLE in the completion and analyzing its implications for hypothesis testing and confidence intervals.

Contribution

It introduces a practical algorithm for MLE estimation in the Barndorff-Nielsen completion and explores the impact on likelihood ratio tests and confidence intervals.

Findings

Likelihood ratio tests are largely unaffected when the MLE lies in the completion.

Confidence intervals are significantly altered when the MLE is in the completion.

A new one-sided confidence interval for the natural parameter is proposed.

Abstract

When in a full exponential family the maximum likelihood estimate (MLE) does not exist, the MLE may exist in the Barndorff-Nielsen completion of the family. We propose a practical algorithm for finding the MLE in the completion based on repeated linear programming using the R contributed package rcdd and illustrate it with two generalized linear model examples. When the MLE for the null hypothesis lies in the completion, likelihood ratio tests of model comparison are almost unchanged from the usual case. Only the degrees of freedom need to be adjusted. When the MLE lies in the completion, confidence intervals are changed much more from the usual case. The MLE of the natural parameter can be thought of as having gone to infinity in a certain direction, which we call a generic direction of recession. We propose a new one-sided confidence interval which says how close to infinity the natural parameter may be. This maps to one-sided confidence intervals for mean values showing how close to the boundary of their support they may be.