Rocky Mountain Journal of Mathematics

When fourth moments are enough

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

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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

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

Keywords
Binomial distribution estimation concentration inequalities machine learning

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


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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.