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What is the best (smallest) bound found so far for the error when using the Poisson approximation to the Binomial distribution? And for which values of $n$ and $p$ is the error smaller than the normal approximation?

I couldn't find anything online.

As a side note, I do have found a bound for the normal approximation (although I'm not sure it's the best)

Normal approximation bound I have found (accordind to this SE question : Berry-Esseen bound for binomial distribution)

$$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}, \quad C \leq .4215$$

Julien__
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  • You could probably improve your normal approximation with a continuity correction. Presumably $\text{Pois}(np)$ is your Poisson approximation to $\text{Bin(n,p)}$ - I doubt it is optimal on this measure of error – Henry Jun 01 '18 at 11:39
  • @Henry, I haven't found any theoretical bound for the continuity correction, only empirical evidence. Do you know any? – Julien__ Jun 01 '18 at 11:42

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