Talk:Panjer recursion

Merger proposal

Copied from Wikipedia talk:WikiProject Statistics#Another article to consider:

I have come across the article (a,b,0) class of distributions which is untouched for some time and has essentailly no articles linking to it. Questions are: Is there a better name for this? Is it just a special case of something else? Melcombe (talk) 09:04, 16 June 2009 (UTC)

Never heard of it before, but it appears to be the same thing as Panjer recursion#Claim number distribution. Merge? Note that Panjer class already redirects to this section. Google reveals that Panjer defined an (a,b,1) class too, but Wikipedia appears to have nothing on that at present. Qwfp (talk) 10:28, 16 June 2009 (UTC)

Please continue discussion below. Qwfp (talk) 11:02, 16 June 2009 (UTC)

This has prompted me to look in Johnson,Kotz & Kemp's book on Univariate Discrete distributions ... the "(a,b,0) class of distributions" seems to be identical to what they call the "Katz family" due to Katz's work over 1945-65, and which is a special case of a family worked on by Carver 1919-25. Ord's book "Families of Frequency Distributions" covers the general case of Carver and I think identifies the "(a,b,0) class of distributions" as a special case which he calls type III without associating anyone's name with it. It seems there may be enough known about these distributions to have results for moments and estimation in an article on the distribution. What is not clear at prresent is whether the more general form of "Panjer recursion" is different from Carver's family of distributions, which is also given in recursive form, but it seems it must be. Melcombe (talk) 12:39, 16 June 2009 (UTC)

Here are my thoughts. Exam C of preliminary exams of the Society of Actuaries has questions about the (a,b,0) (and by extension (a,b,m) distributions). The SOA uses Loss Models as the main (and almost only) required material for studying Exam C (along with Derivatives Markets which accounts for the financial and simulation questions of the exam). Loss Models is co-authored by Panjer. On the other hand, I'll note that nowhere does (or did) the SOA use any other way of calling it than (a,b,0) (or (a,b,1)...) class of distributions.

I should also note that for quite some time now I've been thinking about adding a section to the article which would be called something like (a,b,m) distributions, which would address the remarks made about (a,b,1) distributions. Basically, an (a,b,1) distribution is simply a distribution of the (a,b,0) family that has its p₀ weight fixed, and the other weights being proportionally adjusted. You might also want to fix p₁, p₂, etc. In those cases, the recursive relationship only starts after the last fixed weight.

The only reason why I have no clear or final opinion on the issue is basically because I've never heard of "Panjer's" thingy or the "Katz family" and other stuff like that, so I can't determine whether having a separate article makes sense or not. However, because the SOA is the governing body for actuarial in the United States, as well as a big part of it in Canada and Mexico, I'd suggest trying to keep in line with what they use, unless other actuarial societies (in the UK or in Australia, for instance) use something else.

Well, those were my two cents. :) Seigneur101 (talk) 13:36, 16 June 2009 (UTC)

All good points. Established names in particular disciplines need to be respected but there need not be separate articles. At present the case of starting from some initial values is not covered and it is not clear how it would be done as there is a need to make the values of p sum to one. In addition it would be good to have some proper context about actuarial uses in an improved article. Melcombe (talk) 11:56, 18 June 2009 (UTC)

On looking at the existing Panjer recursion article, I wonder if there isn't an intrinsic problem with it. It seems to start and end with things related to combining information from several distributions in a way that might make sense. But then there is the stuff about "Claim number distribution" in the middle that may or may not be related but which might be traced to Panjer and which includes a recurive structure but it is not clear that it is the same "recursion" or whether it is in a related context. It looks as if it might just have been shoved in, but it seems to have been there from the start.

The general form of the problem stated at the beginning of the Panjer recursion article seems to be rather different from the ideas behind (a,b,0) class of distributions and even "Claim number distribution", so it might be best to start be separating out the "Claim number distribution" material and then possibly merging that into (a,b,0) class of distributions .

Melcombe (talk) 11:56, 18 June 2009 (UTC)

I've changed the {{merge}} templates to be agnostic about the direction as on second thoughts I agree that merging most of the section "Panjer recursion#Claim number distribution" into "(a,b,0) class of distributions" would make more sense. There seems to be no single generally accepted name for these distributions unfortunately. I noticed that this paper (Hess Liewald, Schmidt 2002) calls any member of the (a, b; k) class a "Panjer distribution". I'm not sure what the best title is, though we can have several redirects, of course. Maybe leave as "(a,b,0) class of distributions" for now, but if it's expanded to cover (a,b,1) etc it might make sense to retitle to e.g. "(a,b;k) class of distributions" or "Panjer class of distributions" ? Qwfp (talk) 16:12, 18 June 2009 (UTC)

There should be a distinct separation between the (a,b,0) class and Panjer's recursion. Note that the recursion provides a way to compute the distribution function of compound distributions when the count (claim number, frequency, etc.) belongs to the (a,b,0) class. However, the compound distribution itself does NOT belong to the (a,b,0) class. If anything, the Panjer Recursion page should be expanded to include the continuous form and some discussion on the stability of the algorithm. This page would make more sense as a discussion on the more general (a,b,m) class, with (a,b,0) linking to it. —Preceding unsigned comment added by 149.173.6.51 (talk) 14:24, 22 June 2009 (UTC)

Closure: given the above, I have moved most of the material on the (a,b,0) class of distributions to that article from Panjer recursion, hopefully leaving just enough to make Panjer recursion understandable. Melcombe (talk) 11:14, 29 September 2010 (UTC)