## 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

Zbl 1357.60001
Full Text:

### 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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