Is there a formal definition of the term "generalized" when used with distributions?

Question

I am interested in peak models that are observed in instrumental analysis. The term "generalized" is commonly used in the context of statistical distributions, referring to a class of distributions that are based on a particular distribution, but are modified to include additional parameters or changes to its shape or properties. For example, we have generalized normal distribution and normal distribution.

(i) Does the term "generalized" have a more formal meaning than the one given above? (ii) Is there a specific protocol in statistics to generalize a simpler distribution to a more complex one? By that I mean how do we add another shape parameter to an existing distribution?

score 3 · Accepted Answer · answered May 15 '23 at 10:27

3

A generalized distribution is just more general than what it generalizes. No more, no less. I haven't encountered rules about its usage beyond that self-evident limitation.

A distribution can be generalized in different ways, and conversely a distribution may be a particular case of various generalized distributions.

Nor is there a rule about whether the word "generalized" appears at all. Thus gamma distributions include exponential distributions, but so do generalized Pareto distributions.

Despite its other limitations, I have found the Wikipedia articles on particular distributions generally excellent, and they and the literature generally are replete with comments about particular families of distribution and yet wider families to which they belong.

The matter is muddied further by whether particular distributions are limiting cases of a generalized family, and not just special cases. Thus some normal distributions are limiting cases of gamma distributions.

There is a mass of small print here. Sometimes it is clearer in one parameterisation rather than another what generalizations are possible and indeed useful or congenial.

answered May 15 '23 at 10:27

Nick Cox

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Thanks for the expansion, could add a reference/ or names of methods for generalizing a distribution? For example, I think convolution must be one of them to add another shape parameter. – AChem May 15 '23 at 12:19
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The possibilities are endless. It all depends on where you want to start and where you want to go. You might as well ask similar questions about generalizing anything. – whuber May 15 '23 at 14:49
I've focused on the query about terminology, namely the title question which is also in effect (i). (ii) strikes me as another question which I have not attempted, not least because it is much larger and quite a different flavour. – Nick Cox May 16 '23 at 07:14
I feel I have asked a spiritual sibling of this question about non-central distributions. – Alexis May 18 '23 at 00:17
@Alexis It's not surprising that the more specific question gets a much better answer than I could give. – Nick Cox May 18 '23 at 11:17

Is there a formal definition of the term "generalized" when used with distributions?

1 Answers1