When Fourth Moments Are Enough

This note concerns a somewhat innocent question motivated by an observation concerning the use of Chebyshev bounds on sample estimates of $p$ in the binomial distribution with parameters $n,p$. Namely, what moment order produces the best Chebyshev estimate of $p$? If $S_n(p)$ has a binomial distribution with parameters $n,p$, there it is readily observed that ${\rm argmax}_{0\le p\le 1}{\mathbb E}S_n^2(p) = {\rm argmax}_{0\le p\le 1}np(1-p) = \frac12,$ and ${\mathbb E}S_n^2(\frac12) = \frac{n}{4}$. Rabi Bhattacharya observed that while the second moment Chebyshev sample size for a $95\%$ confidence estimate within $\pm 5$ percentage points is $n = 2000$, the fourth moment yields the substantially reduced polling requirement of $n = 775$. Why stop at fourth moment? Is the argmax achieved at $p = \frac12$ for higher order moments and, if so, does it help, and compute $\mathbb{E}S_n^{2m}(\frac12)$? As captured by the title of this note, answers to these questions lead to a simple rule of thumb for best choice of moments in terms of an effective sample size for Chebyshev concentration inequalities.