Is there any approximation for the PDF and CDF of a linear combination (weighted sum) of more than 3 correlated (non-independent) chi-square random variables?
The variance of all the variables is the same.
The covariance between all the variables is the same, which is half the variance.
Thus, the covariance matrix has the following shape:
| s s/2 s/2 s/2 |
SIGMA = | s/2 s s/2 s/2 |
| s/2 s/2 s s/2 |
| s/2 s/2 s/2 s |
Sorry for the terrible ASCII "art".
Dear Taimur,
Here are links and attached files in subject.
- distributions - sum of noncentral Chi-square random variables - Cross ...
stats.stackexchange.com/.../sum-of-noncentral-chi-sq...
- A comparison of efficient approximations for a weighted sum of chi ...
link.springer.com/article/10.1007/s11222-015-9583-4
- Distribution of a sum of weighted noncentral chi-square variables ...
link.springer.com/article/10.1007/BF02595410
- On Some Procedures Based on Fisher's Inverse Chi-Square Statistic
file.scirp.org/Html/3-7401378---16_35095.htm
Best regards
maybe i can help you but need more explanation what you want exactly!
Dear Mohamed and Fatemeh.
Thank you for the help offers! I feel I owe you some comments on this issue.
At the end, I managed to solve the problem by first applying spectral decomposition on the quadratic form. This is done by finding the eigenvalues of $\Sigma A$, which become the weights on a linear combination of standard uncorrelated chi-square variables. The procedure is described in detail in page 28 of the attached publication.
Once again, thank you.
Kind regards.
Article Quadratic Forms in Random Variables
Dear Mohammad,
May I suggest that you take a look at the paper in which we used this technique?
S. Liu, T. Rabuske, J. Paramesh, L. Pileggi and J. Fernandes, "Analysis and Background Self-Calibration of Comparator Offset in Loop-Unrolled SAR ADCs," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 65, no. 2, pp. 458-470, Feb. 2018.
If you're not interested in the application part, you may just jump to page 461.
If I can be of assistance, please do not hesitate to contact me.
Kind regards,
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