## When can you describe a population and its component subpopulations with the same parametric family of distributions?

I believe that it is often the case that you are trying to select the best probability distribution to use to describe some phenomenon you are studying, and you have data not only for a population, but for sub-populations of that population. I am trying to better understand the constraints that you place on your choice of distribution if you want to (correctly) use the same parametric family of distributions for the sub-populations as for the population as a whole. My interest is primarily in the distribution of income, wealth, and consumption, e.g. comparing state-level distributions to one another, to regional distributions, and to the national distribution. But I believe the problem to be common to many problems and disciplines.

Suppose that, for all X and Y such that both are populations drawn from the a specified parametric family of distributions f(θ), where one or more selected elements of θ differ between the two, a merged super-population Z is also a member of the parametric family f for some value of θ. A trivial example would be two uniform distributions defined over the same interval.

I am looking for all the examples I can find of distributions that have this property. Anybody know of any? Does anyone know if this property, closure of a parametric family under merger of populations, has a standard or conventional name?

Also, if there is any literature on this property and the distributions that posses it, I would be very grateful for pointers to it.

Note that I am not excluding examples where certain elements of θ must be fixed for the property to hold.

I am looking for all the examples I can find, but for my application I am especially interested in continuous distributions over the positive real line.

Any examples, theory, or citations anyone could offer would be most appreciated.

Warmest regards, Andrew

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There are so many possible answers and the subject is so large!

When the parametric family of distributions is some exponential family, there is this recent and very elegant paper:

Although it is focused on studying exponential random graph models, which are commonly used in modeling social networks (and other network models), the framework is quite general.

Edit 1: If the question is about which parametric families are closed under taking mixtures, then the following references are relevant

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 This is a very interesting paper. Thank you so much for bringing it to my attention. But I think my question differs. I am not asking whether, e.g., the parameters of the PDF of income in a state can be used to represent those of the nation, as an estimator. Nor am I asking whether the parameters of the nation can be decomposed into a function of the parameters of the states' distributions. Instead, I am asking whether there are parametric families s.t., if all the state PDFs are drawn from that family with varying parameters, the national PDF will be from the same family for some parameters. – andrewH Sep 23 at 8:04 Hi Andrew. Could you reformulate the question more mathematically? According to your comment, this is a different problem from the consistency under sampling one. If you have say $p$ different sub-population, each distributed according to the same parametric family, but possibly with different parameter, then the distribution of the population is a mixture. So, if that is your question, it becomes then which distributions are closed under mixing. – an12 Sep 23 at 8:56 Hi an12! Thanks for your comment. Yes, you are absolutely correct: I am not talking about samples from a single population, but about a larger population that is divided into disjoint sub-populations with different parameters. I called this a merger, but knowing that I should call it a mixture is very helpful in finding the literature. – andrewH Sep 28 at 21:44