From the Berry-Essen theorem I can deduce

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

with $C \le 0.4748$.

My Question: Is there a better estimate for the constant $C$ than the one given above for the special case of the binomial distribution?

Reason for my question: The given inequality for $C$ holds for any standardized sum of any i.i.d random variables. But I am only interested in the case of binomial distributed random variables. From the answer to my question Estimates for the normal approximation of the binomial distribution I know, that I cannot except a better estimation. But I guess, that there is a better estimate of $C$ if one restricts the Berry-Esseen theorem to binomial distributions only. It would be great when you can point me to an article or a textbook with a better estimate of $C$.

Update: I reasked the question on math.stackexchange.com as suggested in the comments. I hope, this is okay.

  • $\begingroup$ This question maybe has better chances of getting good answers on math.stackexchange.com ? $\endgroup$ – Christoph Hanck Jun 16 '15 at 15:49
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    $\begingroup$ @ChristophHanck: I don't know. For my question Estimates for the normal approximation of the binomial distribution I needed to wait more than a week to get an answer. So I thought this site might be a better choice.... $\endgroup$ – Stephan Kulla Jun 16 '15 at 16:18
  • $\begingroup$ Out of interest, why do you say the best constant for the binomial is $\ge 0.40973$? I just guessed what I think is the worst case and it looks like $C = 0.3704$ is OK (I got that from looking at the worst $n = 1$ case which seems to be $p = 0.390958$..) $\endgroup$ – P.Windridge Jun 16 '15 at 18:05
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    $\begingroup$ I mean, sure, there exists a distribution that requires $C\ge 0.40973$ but (perhaps naively) I think the best $C$ for the binomial is $<0.4$. $\endgroup$ – P.Windridge Jun 16 '15 at 18:08
  • $\begingroup$ @P.Windridge You are right! Thanks for pointing me to this mistake. I corrected it... $\endgroup$ – Stephan Kulla Jun 17 '15 at 22:39

Please don't shoot me if this doesn't work (well) or addresses a different problem than you want.

If your goal is to get the best asymptotic approximation of the Binomial, as opposed to getting the best Berry Esseen bound for its own sake, then consider using an Edgeworth Expansion http://projecteuclid.org/download/pdf_1/euclid.lnms/1215468238 . You may find a Computer Algebra System such as MAPLE to be of value in series manipulation.

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    $\begingroup$ Shooting is frowned upon on this forum - it could lead to a stern moderator message or even several days suspension - so you're probably safe enough not mentioning the risk. $\endgroup$ – Glen_b Jun 18 '15 at 1:35

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