Convolution of iid non central Chi square and normal distribution

[mmmly2002]

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.

 

[chiro]

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?

 

[mmmly2002]

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.

 

[chiro]

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.

 

[blue_raver22]

The convolution of two probability distributions, in this case the iid non central chi-square and normal distribution, is a common technique used in statistics to combine two distributions and obtain a new distribution. In this case, the resulting distribution will have characteristics of both the non central chi-square and normal distribution.

To convolute these two distributions, you can use the convolution theorem, which states that the convolution of two distributions is equal to the product of their Fourier transforms. The Fourier transform of a non central chi-square distribution is a gamma distribution, and the Fourier transform of a normal distribution is another normal distribution. Therefore, the convolution of the two distributions will result in a new distribution that can be expressed as a product of a gamma and a normal distribution.

This new distribution is known as the non central chi-square distribution with non-centrality parameter. It is commonly used in statistical modeling and analysis, particularly in the field of experimental design and hypothesis testing. It is also used in many practical applications such as in signal processing, image processing, and finance.

In summary, convolving the iid non central chi-square and normal distribution can provide a powerful tool for analyzing data and making statistical inferences. It is important to note that the resulting distribution may not have a closed-form expression, but can be easily calculated using numerical methods. I hope this helps in your research and understanding of this topic.