I know there are several approaches to evaluate how well a particular distribution fits a given data set, but how could one evaluate which family of distributions to separately fit to multiple data sets.

For example, let's say I have a dozen datasets of varying size generated by different instances of the same kind of process (but I don't know the inner workings of the process); I want to fit each of them with distinct values for the parameters to the same distribution, but don't know which distribution works best overall. What sort of metric would be useful to compare, e.g., fitting a Weibull to each vs. fitting a lognormal to each?

Assuming, of course, that the GOF for either isn't better than the other for all of the datasets; such strictly dominated options would be removed from consideration in advance.

  • $\begingroup$ You have to be careful to distinguish btw larger distribution families such as gamma and Weibull and the subfamilies they contain. If you give a choice btw 'exponential' and 'gamma' with a sample of moderate size, then 'gamma' almost always wins---even if data are generated from 'exponential'---because 'gamma' has one more parameter to use in fudging a good fit. // Also one should have a clear purpose toward modeling in mind; not much advantage just to have a 'name' for the pop dist'n. $\endgroup$ – BruceET Apr 30 at 1:33
  • $\begingroup$ OK, so take it as read that there is a genuine purpose & comparisons should be done between distribution families with the same number of parameters. $\endgroup$ – Ghillie Dhu Apr 30 at 7:55

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