Convolution of iid non central Chi square and normal distribution

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Discussion Overview

The discussion revolves around the convolution of independent identically distributed (iid) non-central chi-square and normal distributions. Participants explore various approaches to perform this convolution, including characteristic functions and moment generating functions (MGFs), while addressing challenges encountered in the process.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant expresses difficulty in convoluting iid non-central chi-square with a normal distribution and seeks assistance.
  • Another participant asks for clarification on the specific challenges faced and inquires about the methods already attempted, suggesting both straight convolution and MGF approaches.
  • A participant describes their method of using characteristic functions for both distributions, multiplying them, and attempting to perform an inverse Fourier transform, but reports being stuck at this stage.
  • Another participant suggests calculating the probability density function (PDF) of the sum of the two variables and recommends using MGFs, assuming independence, to derive the combined PDF.
  • This participant also proposes alternative methods such as term-by-term integration and approximating the PDF using Taylor series expansions, emphasizing the importance of re-normalizing any approximated PDF.

Areas of Agreement / Disagreement

There is no consensus on the best approach to take for the convolution, as participants present different methods and suggestions without agreeing on a single solution.

Contextual Notes

Participants mention various mathematical techniques and assumptions, such as independence of the distributions and the complexity of the analytic distribution, but do not resolve these issues or clarify all assumptions involved.

Who May Find This Useful

Researchers and students working on statistical distributions, convolution techniques, or those interested in the properties of non-central chi-square and normal distributions may find this discussion relevant.

mmmly2002
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Hi, I am doing research and I am stuck at this point I need help to convolute iid non central chi-square with normal distribution.
 
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Hey mmmly2002 and welcome to the forums.

Can you elaborate on what part you are stuck on? Have you set up the convolution equation? What approaches have you tried? Straight convolution? MGF approach?
 
Thank you for your reply...I really appreciate your help. actually the approach that I used is to take the characteristic function for both non central chi square and normal distribution, then multiply both CF. afetr that take the inverse Fourier transform for the result their production. but I could not solve the inverse Fourier transform for their production and I got stuck at this point...

Thanks again for your help.
 
Are you calculated the PDF of the addition of the two variables?

If so what I recommend is to get the MGF by multiplying the two MGF's (assuming they are independent) and then using the characteristic function for your combined MGF to get the PDF.

Also don't rule out using a term by term integration as opposed to doing something analytically.

If the analytic distribution is extremely complicated and can't easily be expressed with the elementary functions, then what you can do is basically look at the order of the expanding taylor series centred about some point and then cut off the series when the error term (in terms of its order) is large enough.

If you want to do strict calculations, then get an approximation with the right error properties over the domain of the PDF and use that.

You should be able to pick enough terms to reduce the order and you can program a computer to calculate the first n terms and throw them in an array.

But if you use an approximated PDF, make sure you "re-normalize" it so that it has the proper properties of a PDF.
 

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