# Berry-Esseen bound for binomial distribution

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.

• This question maybe has better chances of getting good answers on math.stackexchange.com ? – Christoph Hanck Jun 16 '15 at 15:49
• @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.... – Stephan Kulla Jun 16 '15 at 16:18
• 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 $p = 0.390958$..) – P.Windridge Jun 16 '15 at 18:05
• 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 $<0.4$. – P.Windridge Jun 16 '15 at 18:08
• @P.Windridge You are right! Thanks for pointing me to this mistake. I corrected it... – Stephan Kulla Jun 17 '15 at 22:39