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Jennings-Shaffer, Chris, Skinner, Dane R. and Waymire, Edward C. (2018). WHEN FOURTH MOMENTS ARE ENOUGH. Rocky Mt. J. Math., 48 (6). S. 1917 - 1925. TEMPE: ROCKY MT MATH CONSORTIUM. ISSN 1945-3795

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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 <= p <= 1) ESn2 (p) = argmax(0 <= p <= 1) np(1 - p) = 1/2, and ESn2(1/2) = n/4. Bhattacharya [2] observed that, while the second moment Chebyshev sample size for a 95 percent confidence estimate within +/- 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 = 1/2 for higher order moments, and, if so, does it help in computing ESn2m (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.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Jennings-Shaffer, Chris UNSPECIFIED UNSPECIFIED UNSPECIFIED Skinner, Dane R. UNSPECIFIED UNSPECIFIED UNSPECIFIED Waymire, Edward C. UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-201119
DOI:	10.1216/RMJ-2018-48-6-1917
Journal or Publication Title:	Rocky Mt. J. Math.
Volume:	48
Number:	6
Page Range:	S. 1917 - 1925
Date:	2018
Publisher:	ROCKY MT MATH CONSORTIUM
Place of Publication:	TEMPE
ISSN:	1945-3795
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language Mathematics Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/20111