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.
Highlights
Let x1, x2, . . . , xn be n real numbers with first and second order moments sk = (1/n) ∑ni=1 xik, k = 1, 2
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 sk = (1/n) ∑ni=1 xik, k ≥ 3, into account. We demonstrate that this can be done using the socalled Chebyshev-Markov extremal distributions based on the moments of order three and four or equivalently on the knowledge of the skewness and kurtosis of a real sequence
Symmetric sequences from a Cauchy distribution, whose moments do not exist, generate examples for this phenomenon
Summary
We demonstrate that this can be done using the socalled Chebyshev-Markov extremal distributions based on the moments of order three and four or equivalently on the (sample) knowledge of the skewness and (excess) kurtosis of a real sequence. The latter quantities are denoted and defined by. Symmetric sequences from a Cauchy distribution, whose moments do not exist, generate examples for this phenomenon
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
