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I am trying to find out the expression of the CDF of the product of two independent non-central chi distributions (not chi-squared) each with $k=2$ degrees of freedom. As I understand, non-central chi distribution (with k=2) is square root of the sum of squares of two Guassian random variables of different means and variances.

If I assume that the two Guassian variables have same means and variances, then the non-central chi distribution reduces to standard Rayleigh. It turns out that a closed form of CDF of the product of two independent Rayleigh distributions does exist (e.g. p. 56 of this book).

However, I have been unable to find a closed form of CDF of the product of the two independent non-central chi distributions. Is there a good reference that I can look into? Any help would be greatly appreciated.

-ryan

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I don't know whether the closed form c.d.f. is available, but the p.d.f. of this particular distribution has been studied before. See Kotz and Srinivasan's Distribution of product and quotient of Bessel function variates.

In the paper, a random variable $X$ is said to have a non-central chi distribution with 2 d.f. if its p.d.f. takes the form $$f_X(x)=\frac{x}{\sigma^2}\exp\left(-\frac{\beta^2+x^2}{2\sigma^2}\right)I_0\left(\frac{\beta x}{\sigma^2}\right),$$ where $I_\nu(z)$ is the modified Bessel function of the first kind. Let $X_1\sim\chi(\beta_1,\sigma_1^2)$ and $X_2\sim\chi(\beta_2,\sigma_2^2)$ be independent non-central chi r.v.'s with 2 d.f. The authors showed that the p.d.f. of $Y=X_1X_2$ is $$f_Y(y)=\exp\left(-\frac{\beta_1^2}{2\sigma_1^2}-\frac{\beta_2^2}{2\sigma_2^2}\right)\frac{y}{\sigma_1^2\sigma_2^2}\times\sum_{j=0}^\infty\left[\left(\frac{\beta_2}{\sqrt2\sigma_2^2}\right)^{2j}y^{j\sigma_2/\sigma_1}\sum_{i=0}^j\left(\frac{[(\beta_1\sigma_2)/(\beta_2\sigma_1)]^{i}}{i!(j-i)!}\right)^2K_{i-(j/2)}\left(\frac{y}{\sigma_1\sigma_2}\right)\right],$$ where $K_\nu(x)$ is the modified Bessel function of the second kind.

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  • $\begingroup$ Thanks. I have seen the pdf expression. But I have been looking for the CDF. $\endgroup$ – ryan80 Feb 19 '17 at 9:14
  • $\begingroup$ The reference Kotz and Srinivasan actually gives a pdf of the product of non-central chi distribution with identical variances of constituting Guassian variables. But the question is the product of 'generalized non-central chi' distributions where the Guassian variables have different means and variances. $\endgroup$ – ryan80 Mar 26 '17 at 1:36

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