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