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What is the skew normal approximation to Poisson($\lambda$)?

Am I doing this wrong?

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    $\begingroup$ Why do you want such an approximation? I guess you could match first, second & third moments. $\endgroup$ Commented Jul 2, 2014 at 19:23
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    $\begingroup$ The Skew Normal, although it fits the Poisson PDF reasonably well (by moment matching) for $\lambda\gt 10$ or so, does not look like a good choice for an approximation because computing its CDF and inverse CDF require numerical quadrature. There exist much simpler approximations. Johnson, Kotz & Kemp (Univariate Discrete Distributions, 2nd Ed.) give three pages of them. $\endgroup$
    – whuber
    Commented Jul 2, 2014 at 19:45
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    $\begingroup$ Yes, you're doing it wrong. You need to solve for the three skew-normal parameters in terms of (say) the first three moments of the Poisson distribution. Its mean is $\lambda$, its SD is $\sqrt{\lambda}$, and its skewness is $1/\sqrt{\lambda}$ and from these you can solve for the SN parameters whenever $\lambda\gt 1$. But if computing an incomplete Gamma is too much trouble, then computing a skew normal CDF will be worse. $\endgroup$
    – whuber
    Commented Jul 2, 2014 at 20:09
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    $\begingroup$ It sounds like it would be profitable for you to modify your question so that it asks for suggestions about approximating the Poisson CDF by means of simple fast calculations. If you choose to do that, then please also provide some indication of (a) the range of $\lambda$ of interest (Poissons with $\lambda\ll 1$ can be particularly challenging) and (b) the range of values at which the approximation needs to be accurate. (E.g., in the tails? Upper tail only? Body of the distribution?) $\endgroup$
    – whuber
    Commented Jul 2, 2014 at 20:20
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    $\begingroup$ Based on that SO question: are you actually interested in approximating binomial (rather than Poisson) CDFs by simple fast calculations? It might be better to make the new question directly about binomial distribution, as the optimal(?) method could be something different than approximating the CDF of a Poisson approximation. $\endgroup$ Commented Jul 2, 2014 at 20:46

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