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 $C$.
Note: I am aware, that there are better approximations for the binomial distribution than the normal one. See this answer.
I already asked this question on stats.stackexchange.com, but I didn't get an answer there. I hope it is okay to reask the question here.
Best Answer
The paper "On The Bound of Proximity of the Binomial Distribution to the Normal One" - Nagaev, Chebotarev (2010) has improvements to $C$ specifically for the Binomial Distribution.
Theorem 2 on page 3 gathers together the results in the paper to show that $C$ can be taken to be .4215 in general. The paper notes that an Esseen (1956) paper demonstrates that the bound has to be at least $$\frac{\sqrt{10} + 3}{6\sqrt{2\pi}} \approx .409732$$ so they have found a relatively close bound.
However, there are some better bounds stated in the paper for specific situations. When $n \leq 2000$ and $.02 \leq p \leq .5$ the bound can be taken to be $.4096$. When $0 < p < .02$ it can be taken to be $.369$.