5

if $X_1,...,X_n$ are independent random variables with noncentral chi distributions (same $df$ but different $\lambda$),

What is the distribution of $\sum_{i=1}^{n}{X_i}$

Just wondering if it can be represented by any popular distribution(which has its page on Wikipedia). And want to clarify that I am asking about Chi distribution, not Chi-square.

Nika Tsereteliii
  • 51
  • 3
    OP asked it earlier too @wolfies. That time, I and Xi'an closed it on the same ground. However, OP clarified the query is not at all same. So, I and others voted to reopen it. Sadly, the user decided to delete the post and altogether ask the same and thus history being repeated (albeit I am not involved in the review this time). – User1865345 Apr 30 '23 at 06:57
  • 4
    I recognize you're new here, Nika. To effectively use the site, please be aware that it is best, as @whuber noted before, not to delete an unanswered question & then recreate it. You should edit your post instead. Otherwise the whole cycle starts all over again with people asking for clarifications or providing some food for thought w/o having seen that much of that had already happened. – gung - Reinstate Monica Apr 30 '23 at 12:09
  • Yes, understood. Sorry for that. – Nika Tsereteliii Apr 30 '23 at 15:29
  • 2
    "Just wondering if it can be represented by any popular distribution(which has its page on Wikipedia)" Already when all $\lambda = 0$, then there is no well know distribution related to this sum. Is that a sufficient answer, or are you looking for an approximation method as alternative? – Sextus Empiricus Apr 30 '23 at 21:58
  • I want to somehow derive the distribution of the sum and an approximation method would also be very helpful. @SextusEmpiricus – Nika Tsereteliii May 01 '23 at 09:23
  • 1
    @NikaTsereteliii for specific cases there might be an analytical solution. (I tried something simple like the sum of two chi distributed variables with df=2 and got something like $\exp(x/2)\text{erf}(x)$). For other cases it might be better to use some sort of approximation. If some more clear goal is known then this would help to narrow down the question and make it less broad (currently I perceive it as too broad and I would have no idea where to start). – Sextus Empiricus May 01 '23 at 09:50
  • @SextusEmpiricus I am writing BA thesis and with some assumptions, political polarization in my country has the distribution which can be represented by the sum of noncentral chi random variables. Currently, I am just developing theoretical background and don't have any concrete numbers. As I understood deriving the distribution is too cumbersome(at least for me), So I will add some assumptions to simplify the problem. – Nika Tsereteliii May 01 '23 at 12:21
  • 2
    Non-central chi distributions are analytically fairly intractable. Why not adopt a family of distributions with the same general properties but that are much easier to work with? Otherwise, you will have to solve a host of relatively uninteresting problems in mathematics and computing just to make progress with your subject of interest and even then the solutions might not afford much insight. – whuber May 01 '23 at 17:44
  • 2
    "political polarization in my country has the distribution which can be represented by the sum of noncentral chi random variables." The most interesting problem seems to be why and how it is that political polarization can be represented by a sum of noncentral chi random variables. Possibly, when the underlying mechanisms and motivations are explained, then a decent model for that sum can be derived that does not need to deal directly with 'noncentral chi random variables'? Also, do you really need a good model of the distribution or aren't mean and variance (maybe also kurtosis) enough? – Sextus Empiricus May 01 '23 at 19:15
  • Yes, actually I only need mean and variance of the distribution. I want the whole distribution just for the elegance. I though saying that: "political polarization in this period has this popular distribution" would be nicer. My problem is very specific and I will consult statistic's professor in university. I appreciate your concern and help and don't want you to waste time on my very specific problem. Thanks for help though. – Nika Tsereteliii May 02 '23 at 09:27

0 Answers0