3
$\begingroup$

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?

$\endgroup$
0

1 Answer 1

3
$\begingroup$

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.

$\endgroup$
5
  • $\begingroup$ 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. $\endgroup$
    – AChem
    May 15, 2023 at 12:19
  • 1
    $\begingroup$ 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. $\endgroup$
    – whuber
    May 15, 2023 at 14:49
  • $\begingroup$ 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. $\endgroup$
    – Nick Cox
    May 16, 2023 at 7:14
  • $\begingroup$ I feel I have asked a spiritual sibling of this question about non-central distributions. $\endgroup$
    – Alexis
    May 18, 2023 at 0:17
  • $\begingroup$ @Alexis It's not surprising that the more specific question gets a much better answer than I could give. $\endgroup$
    – Nick Cox
    May 18, 2023 at 11:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.