Testing correlation coefficients from two bivarate poisson

Question

I have datasets from two bivariate poisson distributions, $BVP_x(\lambda_1, \lambda_2, \lambda_{12})$, and $BVP_y(\lambda_3, \lambda_4, \lambda_{34})$ respectively.

Now we know the correlation coefficient for these two distributions can be defined as $\rho_x = \frac{\lambda_{12}}{\sqrt{(\lambda_1 + \lambda_{12})(\lambda_2 + \lambda_{12})}}$ and$\rho_y = \frac{\lambda_{34}}{\sqrt{(\lambda_3 + \lambda_{34})(\lambda_4 + \lambda_{34})}}$,

Is there a standard test for testing $H_0 : \rho_x = \rho_y$ vs $H_1 : \rho_x \neq \rho_y$, I understand this doable by likelihood ratio tests, but not sure if there is already existing distribution for testing the parameters.

I know for bivariate normal distribution Fisher's transformation and $z$ test would work. Would that also work for bivariate poisson.

Answer 1

Yes, the Fisher z-transform would be a reasonable test (even if the underlying distributions are non-normal). The reason for this is that the distribution of the correlation coefficient is nearly normal with the transform. Furthermore, this is empirical in nature (as opposed to something derived from a distribution...though I'm certain there has been a proof using the central limit theorem by this point in time).

The reference you might wish to review is: https://www.jstor.org/stable/20156574.

Hope this helps.