3

What is the difference between a class of distributions and a family of distributions?

The class of (a,b,0) distributions is defined as: enter image description here The Binomial, Poisson, Geometric and Negative Binomial meet these criteria.

The exponential family is defined as

enter image description here

Is the classifier "class" reserved for those distributions that satisfy a recursive relationship and "family" reserved for those distributions that can be transformed into a specific functional form (i.e. not a recursive relationship)?

rocinante
  • 661
  • No, "class" is not restricted to recursively defined distributions. – Glen_b May 13 '14 at 10:22
  • Then what is the point of classifying distributions as part of a "class" and as part of a "family"? Don't class and family function as classification systems? – rocinante May 13 '14 at 10:26

1 Answers1

3

Let us take a look at the use of the terms class and family in mathematics. See, e.g., What is the difference between "family" and "set"? and What are the differences between class, set, family, and collection?.

To summarize, a class of distributions would be any collection of distributions (i.e., set, if we forget advanced set-theoretical considerations motivated by Russell's paradox telling us that not every collection is a set). On the other hand, a family of distributions is an indexed collection of distributions.

The Exponential family Wikipedia article states that "The term exponential class is sometimes used in place of 'exponential family'", which is consistent with the aforementioned definitions: any family of distributions is also a class of distributions if we don't emphasize the indexing.

kjetil b halvorsen
  • 77,844
Juho Kokkala
  • 7,913
  • 1
    +1 This 'index' is very loosely used, like in the exponential family it are the sufficient statistic $T(x)$, natural parameter $\eta(\theta)$, and base measure $h(x)$, that define a specific member in the set. Or in the Pearson distributions it are the values of parameters in a differential equation. Yet, in all these cases the family has every member defined. Sets of distributions that do not have this and would be called a class are for example the class of all discrete distributions, the class of all leptokurtic distributions, the class of all distributions with domain on $x \in [0,1]$ etc. – Sextus Empiricus Mar 24 '24 at 21:06
  • @SextusEmpiricus: So the exponential family is not a family of distributions, but a family of families of distributions (the family of Poisson distributions, the family of Gaussian distributions, &c.)? – Scortchi - Reinstate Monica Mar 25 '24 at 14:57
  • 1
    @Scortchi-ReinstateMonica I agree now that 'the exponential family' is a family of families. https://stats.stackexchange.com/questions/452280/is-there-just-one-exponential-family-or-are-there-many-exponential-families If it would be considered as a family of single densities/distributions, then it would encompass the entire space of potential distributions. For every potential distribution $f(x)$ there are some $h,T,\eta,\theta$ that match. – Sextus Empiricus Mar 26 '24 at 16:49