Rocky Mountain Journal of Mathematics

When fourth moments are enough

Chris Jennings-Shaffer, Dane R. Skinner, and Edward C. Waymire

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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 $S_n(p)$ has a binomial distribution with parameters $n$, $p$, then it is readily observed that ${argmax }_{0\le p\le 1}{\mathbb E}S_n^2(p) = {argmax }_{0\le p\le 1}np(1-p)= \frac{1}{2}$, and ${\mathbb E}S_n^2(\frac{1}{2}) = \frac{n}{4}$. Bhattacharya observed that, while the second moment Chebyshev sample size for a 95 percent 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 the fourth moment? Is the argmax achieved at $p = \frac{1}{2}$ for higher order moments, and, if so, does it help in computing $\mathbb {E}S_n^{2m}(\frac{1}{2})$? As captured by the title of this note, answers to these questions lead to a simple rule of thumb for the best choice of moments in terms of an effective sample size for Chebyshev concentration inequalities.

Article information

Rocky Mountain J. Math., Volume 48, Number 6 (2018), 1917-1924.

First available in Project Euclid: 24 November 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx} 62D05: Sampling theory, sample surveys

Binomial distribution estimation concentration inequalities machine learning


Jennings-Shaffer, Chris; Skinner, Dane R.; Waymire, Edward C. When fourth moments are enough. Rocky Mountain J. Math. 48 (2018), no. 6, 1917--1924. doi:10.1216/RMJ-2018-48-6-1917.

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