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

Am I doing this wrong?

Museful
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    Why do you want such an approximation? I guess you could match first, second & third moments. – kjetil b halvorsen Jul 02 '14 at 19:23
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    Hi Museful and welcome to the site. Could you please expand your question a little? Why do you need such an approximation? What questions are you trying to solve and with what data? What have you tried? – COOLSerdash Jul 02 '14 at 19:29
  • @kjetilbhalvorsen I need to implement a rough-and-very-fast software routine for Poisson CDF and am exploring different possibilities of which this is one. Thanks for the hint. – Museful Jul 02 '14 at 19:34
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    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. – whuber Jul 02 '14 at 19:45
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    The exact Poisson CDF can be expressed via the Gamma function, se wikipedia article. Isnt that fast enough? – kjetil b halvorsen Jul 02 '14 at 19:47
  • @kjetilbhalvorsen That is my base implementation but it needs the *incomplete* gamma function which requires an iterative converging algorithm. – Museful Jul 02 '14 at 20:07
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    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. – whuber Jul 02 '14 at 20:09
  • @whuber Yes. For some reason I had the wrong impression after skimming wikipedia too fast that SN CDF could be done with erf. Thanks for your help. Unfortunately I don't have enough rep to upvote comments. – Museful Jul 02 '14 at 20:12
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    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?) – whuber Jul 02 '14 at 20:20
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    @whuber Thanks. After absorbing all the input above I will probably post that as a separate question as it is a completely different question, and in case someone is answering this. People in this place seem to be [comparatively very patient with newbies](http://stackoverflow.com/questions/24460776/fast-incomplete-gamma-function). – Museful Jul 02 '14 at 20:39
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    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. – Juho Kokkala Jul 02 '14 at 20:46

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