# Estimates for the normal approximation of the binomial distribution

I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of

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

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: Are there better estimates for the normal approximation of the binomial distribution?

The Berry-Esseen theorem is quite general because it can be applied to each sum of i.i.d random variables. So I guess there are better estimates for the special case of the binomial distribution...

By Chebyshev's inequality, a positive proportion of the mass of the binomial distribution is located between $np-\sqrt{npq}$ and $np+\sqrt{npq}$. Together with the pigeonhole principle, this implies that a single value in this interval is taken on with probability proportional to $\frac{1}{\sqrt{npq}}$ (there are much more precise results than this; see for example Steven Dunbar's notes on Local Limit Theorems), meaning that the CDF of the binomial distribution has jumps of this size.
On the other hand, the normal distribution is continuous. So on one side or the other of the jump, the approximation error must be at least $\frac{C'}{\sqrt{npq}}$, which matches the Berry-Esseen bound up to a constant factor ($p^2+q^2$ is always between $1/2$ and $1$).