When Fourth Moments Are Enough

TL;DR

This paper investigates the optimal moment order for Chebyshev bounds on binomial sample estimates, showing that higher moments like the fourth can significantly reduce required sample sizes for confidence estimates.

Contribution

It provides a theoretical analysis of moments beyond the second for Chebyshev bounds, offering a simple rule of thumb for selecting moments to improve sample efficiency.

Findings

Fourth moments reduce sample size requirements compared to second moments.

Higher moments achieve similar confidence with fewer samples.

A simple rule of thumb guides optimal moment choice for Chebyshev bounds.

Abstract

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.