## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 3 (2009), 259-289.

### Likelihood inference in exponential families and directions of recession

#### 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 three 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.

#### Article information

**Source**

Electron. J. Statist., Volume 3 (2009), 259-289.

**Dates**

First available in Project Euclid: 14 April 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1239716414

**Digital Object Identifier**

doi:10.1214/08-EJS349

**Mathematical Reviews number (MathSciNet)**

MR2495839

**Zentralblatt MATH identifier**

1326.62070

**Subjects**

Primary: 62F99: None of the above, but in this section

Secondary: 52B55: Computational aspects related to convexity {For computational geometry and algorithms, see 68Q25, 68U05; for numerical algorithms, see 65Yxx} [See also 68Uxx]

**Keywords**

exponential family existence of maximum likelihood estimate Barndorff-Nielsen completion

#### Citation

Geyer, Charles J. Likelihood inference in exponential families and directions of recession. Electron. J. Statist. 3 (2009), 259--289. doi:10.1214/08-EJS349. https://projecteuclid.org/euclid.ejs/1239716414