Difficulty with summation of non-central chi-squared random variables

[Jeff.Nevington]

Hi,
I am struggling trying to find the (equation of the) pdf of the sum of (what I believe to be) two non-central chi-squared random variables.

The formula given on wikipedia (http://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution) shows that the random variable associated with a sum of squared normal variables normalized by their variances is "non-central chi squared". The problem I am having is that I do not want to normalize my normal random variables by their variances; I just want to add together the squared values. For example, if X1 and X2 are normally distributed random variables with non identical means and non identical variances, what is the pdf of X1^2+X2^2?

So far my solution has been to calculate the individual pdfs of X1^2 and X2^2 and perform a numerical convolution to find the pdf X1^2+X2^2 - it just seems silly not to have an analytical solution if one has already been created for this exact problem.

Thanks,
Jeff.

 

[mathman]

Are you familiar with using characteristic functions (c.f.'s)? If you have the pdf's of the squares, you can take their Fourier transforms (these are the c.f.'s) , multiply them, and take the back transform of the product. This is the pdf that you want.

 

[Jeff.Nevington]

Thanks for the reply. I did actually reply last night but the post seems to have disappeared... Anyway, I looked up the characteristic function of the non-central chi-squared distribution as you suggested. When multiplying these characteristic functions together they simplify to the exact same form as the single function (if the variances are equal). In this case getting back to the pdf from the new characteristic function is easy as the derivation has been done for me. However when the variances are different, getting back to the pdf looks like it would require performing some difficult integrals and as my integral solver doesn't seem to be able to do it, I don't fancy my chances. I am happy enough to come to the conclusion that it is a very difficult problem and settle with a numerical solution - I just expected that somewhere I would find clarification that it is very difficult. What seems odd is that the formula for finding the sum of non-central chi squared variables with normalized variance is easy enough to find but there is no mention of the significance of normalizing the variance. Surely in most situations when wanting to add R.Vs with different variances, normalizing the variances would severely skew the pdf.

 

[kdl05]

Hi Jeff, others,

Did you ultimately find a way to solve this problem? I'm currently very much interested in finding the solution, and am having no luck finding an analytic solution. I'm not familiar with the numerical methods discussed in this thread. Could someone offer a more detailed description for how to go about solving this problem numerically, and/or provide some references.

As stated in the original question, the difficulty I'm having is when the two normally distributed rv's X1 and X2 have different means and different variances, and I would like to NOT normalize by the variance, as is done for a "non-central chi squared" distribution.

Thanks a lot!

 

[blue_raver22]

Dear Jeff,

I understand your struggle with finding the pdf for the sum of non-central chi-squared random variables. This is a common challenge for many scientists and researchers in the field of statistics. The formula given on Wikipedia is indeed for the normalized non-central chi-squared distribution, which may not be applicable to your specific scenario.

In order to find the pdf for the sum of squared normal random variables, X1^2+X2^2, you can use the moment generating function (MGF) approach. The MGF of the sum of two independent random variables is the product of their individual MGFs. In this case, the MGF of X1^2+X2^2 can be calculated by taking the product of the MGFs of X1^2 and X2^2. This will give you an analytical solution for the pdf of X1^2+X2^2.

Alternatively, you can also use the characteristic function approach to find the pdf. The characteristic function of the sum of two independent random variables is the product of their individual characteristic functions. Again, this will give you an analytical solution for the pdf of X1^2+X2^2.

I hope this helps in your search for an analytical solution. Keep in mind that in some cases, numerical convolution may be the only option, but it is always worth exploring analytical solutions first. Best of luck in your research.

Sincerely,