An Improved Laguerre-Samuelson Inequality of Chebyshev-Markov Type

Journal of Optimization, Aug 2014

The Chebyshev-Markov extremal distributions by known moments to order four are used to improve the Laguerre-Samuelson inequality for finite real sequences. In general, the refined bound depends not only on the sample size but also on the sample skewness and kurtosis. Numerical illustrations suggest that the refined inequality can almost be attained for randomly distributed completely symmetric sequences from a Cauchy distribution.

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An Improved Laguerre-Samuelson Inequality of Chebyshev-Markov Type

An Improved Laguerre-Samuelson Inequality of Chebyshev-Markov Type Werner Hürlimann Swiss Mathematical Society, University of Fribourg, 1700 Fribourg, Switzerland Received 2 May 2014; Accepted 29 July 2014; Published 12 August 2014 Academic Editor: Cheng-Hong Yang Copyright © 2014 Werner Hürlimann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The Chebyshev-Markov extremal distributions by known moments to order four are used to improve the Laguerre-Samuelson inequality for finite real sequences. In general, the refined bound depends not only on the sample size but also on the sample skewness and kurtosis. Numerical illustrations suggest that the refined inequality can almost be attained for randomly distributed completely symmetric sequences from a Cauchy distribution. 1. Introduction Let <glyph.data ascent="3473" descent="-2876" horiz-adv-x="559" vert-adv-y="559"></glyph.data><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="559" vert-adv-y="559"></glyph.data><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="559" vert-adv-y="559"></glyph.data><glyph.data ascent="3443" descent="-2856" horiz-adv-x="512" vert-adv-y="512"></glyph.data> be <glyph.data ascent="3473" descent="-2876" horiz-adv-x="502" vert-adv-y="502"></glyph.data> real numbers with first and second order moments <glyph.data ascent="3473" descent="-2876" horiz-adv-x="375" vert-adv-y="375"></glyph.data><glyph.data ascent="3443" descent="-2856" horiz-adv-x="509" vert-adv-y="509"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="412" vert-adv-y="412"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="502" vert-adv-y="502"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="750" vert-adv-y="750"></glyph.data><glyph.data ascent="3443" descent="-2856" horiz-adv-x="512" vert-adv-y="512"></glyph.data><glyph.data ascent="3443" descent="-2856" horiz-adv-x="280" vert-adv-y="280"></glyph.data><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="559" vert-adv-y="559"></glyph.data><glyph.data ascent="3443" descent="-2856" horiz-adv-x="509" vert-adv-y="509"></glyph.data><glyph.data ascent="3443" descent="-2856" horiz-adv-x="280" vert-adv-y="280"></glyph.data>, <glyph.data ascent="3473" descent="-2876" horiz-adv-x="503" vert-adv-y="503"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data>. The Laguerre-Samuelson inequality (see Jensen and Styan [1] and Samuelson [2]) asserts that for a sample of size <glyph.data ascent="3473" descent="-2876" horiz-adv-x="502" vert-adv-y="502"></glyph.data> no observation lies more than <glyph.data ascent="3473" descent="-2876" horiz-adv-x="764" vert-adv-y="764"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="502" vert-adv-y="502"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data> standard deviation away from the arithmetic mean; that is, Experiments with random samples generated from various distributions on the real line suggest that there is considerable room for improvement if one takes higher order moments <glyph.data ascent="3473" descent="-2876" horiz-adv-x="375" vert (...truncated)


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Werner Hürlimann. An Improved Laguerre-Samuelson Inequality of Chebyshev-Markov Type, Journal of Optimization, 2014, 2014, DOI: 10.1155/2014/832123