I'm trying to find the motivation behind the extended form of the exponential family of distributions in the fundamental paper on GLM by Nelder and Wedderburn (Generalized Linear Models, J. R. Statist. Soc. A (1972), 135, Part 3, p. 370).

In section 1.1 they simply jump right in, and state that they will be dealing with distributions of the form

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But how did they come up with the phi-dependence in this form? What was the motivation for introducing alpha(phi) and beta(phi,z)? Did they just stare at a bunch of distributions that contain a phi-like parameter, and then infer this form? If so, can someone point me to an exhaustive list of the distributions they stared at? :) Or did they use some other considerations for introducing alpha(phi) and beta(phi,z)?

Any insights would be greatly appreciated!

  • 2
    $\begingroup$ In almost every book this distribution family is described as 'exponential family - EF'. The truth is that this isn't EF but its special case called 'Natural exponential family - NEF' . Maybe this could help you. en.wikipedia.org/wiki/Natural_exponential_family $\endgroup$
    – Maju116
    Jan 28 '16 at 12:42
  • $\begingroup$ Thank you, that is very helpful. Odd that the literature doesn't point back to NEF as the foundation of GLM. $\endgroup$ Jan 30 '16 at 20:30

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