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Approximation of skellam distribution by a Gaussian one

  1. Jul 7, 2011 #1
    Hi, everybody

    Let [itex] n_1 [/itex] ~ Poisson ([itex] \lambda_1 [/itex]) and [itex] n_2 [/itex] ~ Poisson ([itex]\lambda_2[/itex]).
    Now define [itex] n=n_1-n_2 [/itex]. We know [itex] n [/itex] has "Skellam distribution" with mean [itex]\lambda_1-\lambda_2[/itex] and variance [itex] \lambda_1+\lambda_2[/itex], which is not easy to deal with.
    I want to find the [itex] Pr(n \geq 0) [/itex]. Is it possible to find a good approximation for the above probability by employing an approximated "Gaussian distribution"? If "Gaussian" is not a good candidate, which distribution can I replace it with?
     
  2. jcsd
  3. Jul 7, 2011 #2
    If at least one of the lambdas is large, the Gaussian with the same mean and variance will be a good approximation.
     
  4. Jul 7, 2011 #3
    But it is not always the case. I want to deal with the more general cases.
     
  5. Jul 7, 2011 #4
    Yes, I know. But for small lambda, I don't think there's any simpler approximation. Of course, you could in that case just truncate the distributions. If the mean is small, the probably of large n is vanishingly small, so it won't introduce much inaccuracy to leave them out.
     
  6. Jul 7, 2011 #5
    To approximate one distribution with another use maximum likelihood, i.e. maximize
    [tex]E[\log(f(X;t))[/tex]
    wrt the parameter vector t, where f is the pdf or pmf of the approximating distribution. E.g. solving for the normal distribution we get [itex]\mu=E[X][/itex] and [itex]\sigma^2=E[X^2]-E[X]^2[/itex].
     
  7. Jul 8, 2011 #6
    Could you please explain it in more details.
     
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