5
$\begingroup$

How would I simulate observations from a bivariate Poisson distribution such that they have a nonzero covariance? The hint I was given is that I need to use the fact that the sum of two Poisson random variables is also Poisson.

$\endgroup$
  • 1
    $\begingroup$ See stats.stackexchange.com/questions/108705. Both answers there show how values can be simulated by means of three independent Poisson variates (although they do not explicitly point that out). $\endgroup$ – whuber Feb 21 '15 at 20:25
  • 2
    $\begingroup$ Please see the self-study tag wiki. The hint you already have is a complete giveaway; I'm not sure there's much of anything between the hint you already have and telling you exactly which Poissons to add ... i.e. a complete solution. Oh, wait, there's one -- start with more than two independent Poissons. $\endgroup$ – Glen_b -Reinstate Monica Feb 22 '15 at 9:53
8
$\begingroup$

Since you do not impose any constraint on the joint distribution, any copula structure gives you a solution. For instance,

  1. take a bivariate normal $$(x_1,x_2)\sim\mathcal{N}_2\left((0,0),\left[\matrix{1 &\rho\\\rho &1}\right]\right)$$generation;
  2. turn $(x_1,x_2)$ in correlated uniforms as $$(u_1,u_2)=(\Phi(x_1),\Phi(x_2))$$where $\Phi(\cdot)$ is the normal CDF;
  3. derive two Poisson variates $(n_1,n_2)$ with parameters $\lambda_1$ and $\lambda_2$ from $(u_1,u_2)$ by inverting the Poisson CDF.
$\endgroup$
  • $\begingroup$ +1 but could you maybe also add an introductory reference to the copula link? It's a wikipedia stub at present. $\endgroup$ – conjugateprior Feb 22 '15 at 10:22
  • $\begingroup$ @conjugateprior: thank you, the link was damaged and I just corrected it to the entire Wikipedia page. $\endgroup$ – Xi'an Feb 22 '15 at 10:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.