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When fourth moments are enough. (English) Zbl 1404.60022

Summary: 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 \(\mathrm{argmax}_{0\leq p\leq 1}{\mathbb E}S_n^2(p)=\mathrm{argmax}_{0\leq p\leq 1}np(1-p)= \frac{1}{2}\), and \({\mathbb E}S_n^2(\frac{1}{2})=\frac{n}{4}\). In [A basic course in probability theory. 2nd edition. Cham: Springer (2016; Zbl 1357.60001)], R. Bhattacharya and E. C. Waymire 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.

MSC:

60C05 Combinatorial probability
60F05 Central limit and other weak theorems

Citations:

Zbl 1357.60001
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References:

[1] R.N. Bhattacharya and E.C. Waymire, A basic course in probability theory, Universitext, Springer, New York, 2016. · Zbl 1357.60001
[2] J. Duchi, M.J. Wainwright and M.I. Jordan, Local privacy and minimax bounds: Sharp rates for probability estimation, in Advances in neural information processing systems, (2013), 1529–1537.
[3] D. Skinner, Concentration of measure inequalities, Master of Science thesis, Oregon State University, Corvallis, 2017.
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