When Fourth Moments Are Enough

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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 Sₙ(p) has a binomial distribution with parameters n,p, there it is readily observed that argmax₀≤ ₚ≤ ₁ESₙ²(p) = argmax₀≤ ₚ≤ ₁np(1-p) = 12, and ESₙ²(12) = n/4. Rabi Bhattacharya observed that while the second moment Chebyshev sample size for a 95% confidence estimate within ± 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 = 12 for higher order moments and, if so, does it help, and compute ESₙ²ᵐ(12)? 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.