Double exponential families and their use in generalized linear regression. (English) Zbl 0611.62072

Efron, Bradley

J. Am. Stat. Assoc. 81, 709-721 (1986).

The author investigates the double exponential family obtained by adding a second parameter to an ordinary one-parameter exponential family. The new parameter varies the dispersion of the family without changing the mean. The family enjoys, at least approximately, some of the useful properties of the \(N(\mu,\sigma^ 2)\) family.
The theory is applied to two examples – a logistic regression and a large two-way contingency table. The paper also concerns a class of regression families that allow the statistician to model over-dispersion while carrying out the usual regression analyses for the mean as a function of the predictors.
The paper finishes with two examples which show the potential of the double exponential families for “robustifying” standard logistic and Poisson regression analyses.

Reviewer: Wang Songgui

MSC:

62J05	Linear regression; mixed models
62J99	Linear inference, regression
62J12	Generalized linear models (logistic models)
62H17	Contingency tables
62E10	Characterization and structure theory of statistical distributions

Keywords:

generalized linear models; double exponential family; dispersion; logistic regression; large two-way contingency table; over-dispersion; robustifying; Poisson regression analyses