## Rocky Mountain Journal of Mathematics

### When fourth moments are enough

#### 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 $n$, $p$, 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

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

Dates
First available in Project Euclid: 24 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1543028445

Digital Object Identifier
doi:10.1216/RMJ-2018-48-6-1917

Mathematical Reviews number (MathSciNet)
MR3879309

Zentralblatt MATH identifier
06987232

#### Citation

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. https://projecteuclid.org/euclid.rmjm/1543028445

#### References

• R.N. Bhattacharya, personal communication, 2016.
• R.N. Bhattacharya and E.C. Waymire, A basic course in probability theory, Universitext, Springer, New York, 2016.
• 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.
• D. Skinner, Concentration of measure inequalities, Master of Science thesis, Oregon State University, Corvallis, 2017.