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Suppose you have two instances of a distribution that are parameterized differently, and for one of them a certain restriction on the parameter values of the pdf or CDF results (perhaps after some algebraic manipulation) in a closed-form expression for the pdf or CDF (respectively) of another named distribution with one fewer parameters, as a special case.

Is it necessarily the case that the second parameterization also has a closed form of the same named distribution as a special case?

Take, for example, the scale and rate form of the gamma distribution. On one hand, these are merely alternative parameterizations of the same distribution, so it would seem that any distributions that nest in one must also nest in the other. On the other hand, I am not sure the limiting distributions are the same. Take the gamma distribution as an example. The rate parameterization and the scale parameterization seem to me to be different enough that distributions that are accessible only as limits might be different.

I am actually hoping that the answer is no: that the limiting distributions of a distribution are independent of parameterization.

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  • $\begingroup$ This looks like an attempt to reformulate stats.stackexchange.com/questions/332797, but it's not entirely clear what you're trying to ask. $\endgroup$ – whuber Mar 11 '18 at 15:59
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    $\begingroup$ I don't think that the questions are the same, but I do think that your answer to that question goes a long way toward answering this one. $\endgroup$ – andrewH Mar 13 '18 at 6:50
  • $\begingroup$ Thank you. Your question about limiting distributions--although not asked--is almost answered at stats.stackexchange.com/questions/320746, which shows that the parameterization is just a way of describing a family of distributions. The (rather technical) part of the question it does not even address is whether there would be two distinct topologies on a parametric family. Since the topology is determined by the relative likelihoods, the answer to that is obviously no. $\endgroup$ – whuber Mar 13 '18 at 14:15
  • $\begingroup$ Well, I can not say that I fully understand your explanation, here or there. They rely on more math (e.g. topology, maybe measure theory or advanced real analysis) than I have. But if I understand your overall thrust, the answer to my original question is no. And since that corresponds to both my intuition and my hopes, I'm prepared to take your statement as a proof by authority. $\endgroup$ – andrewH Mar 13 '18 at 19:18
  • $\begingroup$ But given your discussion of "parametric families" in the posting you reference, I think another way of asking my question is whether two parametric families of the same distribution can enclose different distributions as special cases. I assume the answer is still "no." $\endgroup$ – andrewH Mar 13 '18 at 20:03

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