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Bounds on Sample Variation Measures Based on Majorization

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  • Tarald O. Kvålseth

Abstract

Using majorization theory, upper and lower bounds are derived for different measures of variation as progressively more items of information are available about the sample data. As a convenient starting point, bounds are first established for a one-parameter family of variation measures, which is a generalized mean difference measure of which Gini's mean difference, the standard deviation, and the range are particular cases. While, as pointed out, some of the derived bounds are well known, others do not appear to have been published and are tighter than established bounds. Some 40 different bounds are derived, besides any number of bounds given for the generalized family of variation measures. A number of interesting inequalities are also derived on the basis of some of the bounds. While the bounds have been developed in terms of real-valued sample data generally, the paper concludes with a brief discussion of the bounds for categorical data when the sample data consists of frequencies (counts).

Suggested Citation

  • Tarald O. Kvålseth, 2015. "Bounds on Sample Variation Measures Based on Majorization," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 44(16), pages 3375-3386, August.
    • Handle: RePEc:taf:lstaxx:v:44:y:2015:i:16:p:3375-3386
    DOI: 10.1080/03610926.2013.844252
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