1

I read on wikipedia about the exponential family (https://en.wikipedia.org/wiki/Exponential_family) that: enter image description here , but later on in the same article I read that:

Some distributions are exponential families only if some of their parameters are held fixed. The family of Pareto distributions with a fixed minimum bound xm form an exponential family. The families of binomial and multinomial distributions with fixed number of trials n but unknown probability parameter(s) are exponential families. The family of negative binomial distributions with fixed number of failures (a.k.a. stopping-time parameter) r is an exponential family. However, when any of the above-mentioned fixed parameters are allowed to vary, the resulting family is not an exponential family.

If I want to model something with the Pareto distribution, would I not always know what the absolute lowest value would be, and thus in effect could always treat the Pareto as a exponential family model? Can i make regression models with Pareto distribution, with explanatory variables?

Community
  • 1
Harry Giil
  • 21
  • 1
    "would I not always know what the absolute lowest value would be" -- only you can answer whether you will always know; in some applications at least this is not always known -- sometimes people do estimate that parameter because they don't actually know the value. – Glen_b Dec 03 '17 at 11:24
  • I see, thank you! As for my latter question, I have googled much, but I cannot find a clear answer. Would you happen to know? – Harry Giil Dec 03 '17 at 11:28
  • You could fit a Pareto model with known lower limit by dividing by it (making the lower limit 1), taking logs and fitting an ordinary exponential GLM; I'm not sure that's what you're after though -- it won't be a model for the conditional mean of the Pareto though, since if you exponentiate the exponential fit it would be the (conditional) geometric mean of the Pareto. – Glen_b Dec 03 '17 at 11:52
  • 1
    However, given that the essential information for a GLM would seem to be in the table here I think it may be possible to write the relevant functions to fully implement a Pareto GLM. (I'm not sure; the transformation may still be necessary) – Glen_b Dec 03 '17 at 12:02
  • Looking at the documentation on both functions, that looks like it should do what you want. – Glen_b Dec 03 '17 at 13:25
  • 1
    Distributions in the exponential family have finite bounds on their moments. When those bounds are relaxed or released, exponential assumptions no longer apply and different extreme valued distributions come into play where the moments can be undefined, unbounded or infinite. Examples of such distributions include the Cauchy, which has a single parameter -- the mean -- but the higher moments are undefined or infinite. – user78229 Dec 04 '17 at 13:57
  • See https://stats.stackexchange.com/questions/469099/how-to-implement-a-glm-with-pareto-family and links therein, https://stats.stackexchange.com/questions/245162/is-pareto-distribution-exponential-dispersion-family-and-the-form-is-unique, https://stats.stackexchange.com/questions/115754/pareto-two-tailed-glm-regression – kjetil b halvorsen Jul 22 '23 at 22:47

0 Answers0