Titlebook: Inequalities: Theory of Majorization and Its Applications; Albert W. Marshall,Ingram Olkin,Barry C. Arnold Book 2011Latest edition Springe

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Additional Statistical Applicationsare given. These examples have been chosen in part be- cause the exposition of the statistical background is minimal, so that the main emphasis lies in the use of majorization. There are other ex- amples, not given here, where the statistical material is more involved and/or the majorization does not play as critical a role.

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Doubly Stochastic MatricesAn important tool in the study of majorization is a theorem due to Hardy, Littlewood, and Pólya (1929) which states that for ., ..,. if and only if . = . for some doubly stochastic matrix ..

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Equivalent Conditions for MajorizationIn order to verify that . is majorized by ., the conditions of the definition might be directly checked, but it is sometimes more con-venient to check alternative conditions. The purpose of this chapter is to collect some such alternatives.

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Rearrangements and MajorizationMany inequalities are known that involve real vectors and vectors with the components rearranged. Such inequalities are discussed in Chapter X of Hardy, Littlewood, and Pólya (1934, 1952), but a number of new results have been obtained since that book was written. Particularly 4 noteworthy is the work of Lorentz (1953) and Day (1972).

Coronation

Stochastic MajorizationsA comparison between two random vectors . and . might be called 1 a . if the comparison reduces to the ordinary 2 majorization . in case . and . are degenerate at . and .; i.e.,.

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CONE

PACT

Matrix Theory of matrix theory. These majorizations 7 primarily involve quantities such as the eigenvalues or singular val- 8 ues of matrix sums or products. An integral part of the development 9 of majorization in matrix theory is the extremal representations of 10 Chapter 20.

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Book 2011Latest editionjorization and its Applications” merits strong praise. It is innovative, coherent, well written and, most importantly, a pleasure to read. … This work is a valuable resource!” (Mathematical Reviews). “The authors … present an extremely rich collection of inequalities in a remarkably coherent and uni

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Preservation and Generation of Majorizationarious properties besides ma- jorization, or.is derived from.in a specified manner. These kinds of origins of majorization are summarized in Section B. In Section C, ex- treme vectors (in the ordering of majorization) under constraints are identified, and Section D gives some special results for vectors with integer components.