Supporting Theory and Data Analysis for “ Likelihood Inference in Exponential Families and Directions of Recession ”

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 (BarndorffNielsen, 1978; Brown, 1986; Geyer, 1990). A practical algorithm for finding the MLE in the completion using repeated linear programming was proposed in the author’s unpublished thesis (Geyer, 1990) and used in Geyer and Thompson (1992). Now we propose a slightly different method, also using repeated linear programming with the R contributed package rcdd (Geyer and Meeden, 2008), which makes straightforward the calculation of the MLE in the Barndorff-Nielsen completion for any models satisfying a condition of Brown (1986) and for which some R function can calculate the MLE when it does exist, for example, generalized linear models (GLM) and aster models (Geyer et al., 2007; Geyer, 2008). In this technical report we give details of two GLM examples. Likelihood ratio tests of model comparison are almost unchanged from the usual case. Only the degrees of freedom need be adjusted when the MLE for the null hypothesis lies in the completion rather than the original family. Confidence intervals are changed much more. When the MLE for the natural parameter does not exist, it can be thought of as having gone to infinity in a certain direction, which we call a generic direction of recession. Here we propose a new kind of one-sided confidence interval, not involving asymptotic approximation, for how close to infinity the true unknown natural parameter value may be. This maps to a one-sided confidence interval for the mean value parameter showing how close to the boundary of its support it may be.