Theory of majorization and its applications second. Theory of majorization and its applications, 2nd edition by a. Majorization and dynamics of continuous distributions mdpi. Entropy majorization, thermal adiabatic theorem, and quantum. Nonadditive measure and integral, kluwer academic publishers dordrechtbostonlondon 2001. Measurement of inequality, 2000, in a b atkinson f bourguignon eds. Introduction to global politics, brief third edition, brings together an expert team of international scholars to provide students with a current, engaging, and non u. Rvis, up to scaling, the volume of the lorenz zonoid plus a unit cube of full dimension. Majorization and the lorenz order in statistics, applied probability. Measuring aggregation in parasite populations journal of. Some demographic applications of the univariate and the bivariate forms of the pareto distribution, proceedings of the 45th. Majorization, graph theory, schurconvex function, optimization. A brief introduction, the authors provide a straightforward development and explanation of majorization concepts, addressing historical development of the topics, and providing uptodate coverage of families of lorenz curves. Some demographic applications of the univariate and the bivariate forms of the pareto distribution, proceedings of the 45th session of the international statistical institute, 1, 267268.
Barry c arnold these notes are designed for a one quarter course introducing majorization and the lorenz order. The majorization partial order on finite sets, described here, can be generalized to the lorenz ordering, a partial order on distribution functions. The dashed line is the lorenz curve of a negative binomial distribution with m 3 and k 1. Intergenerational links, taxation, and wealth distribution. Thelorenz zonoidwas suggested as a multivariate generalization of the lorenz curve. We show that the result holds for a stronger criterion of inequality comparison between distributions.
Some recent developments on majorization inequalities in probability and statistics. In mathematics, majorization is a preorder on vectors of real numbers. It is an introduction to the theory of majorization and related notions. It shows students how to analyze global political events using theoretical approaches both mainstream and alternative and emphasizes non state. She is currently working on another young adult novel.
The lorenz order can, thus, be thought of as a useful generalization of the majorization order. Majorization and the lorenz order with applications in. Wallach is the author of six young adult books and has been an editor for thirty years. Entropy majorization, thermal adiabatic theorem, and quantum phase transitions shijian gu department of physics and institute of theoretical physics, the chinese university of hong kong, hong kong, china. Actuarial theory for dependent risks wiley online library.
The lorenz curve, however, can be meaningfully used to compare arbitrary distributions rather than distributions concentrated on n points, as is the case with the majorization partial order. The lorenz curve in economics and econometrics semantic scholar. In this section we give a short account of transfer principles and majorization, and. This paper surveys selected applications of the lorenz curve and related stochastic orders in economics and econometrics, with a bias towards problems in statistical distribution theory. The need for a multivariate version of lorenzs curve and his. Sorry, we are unable to provide the full text but you may find it at the following locations. Multivariate convex orderings, dependence, and stochastic equality volume 35 issue 1 marco scarsini.
The appearance of marshall and olkins 1979 book on inequalities with special emphasis on majorization generated a surge of interest in potential applications of majorization and schur convexity in a broad spectrum of fields. Majorization and the lorenz order with applications in applied. Hardylittlewoodpolyas inequality and a new concept of weak. A brief introduction, lecture notes in statistics, 43, springerverlag. Obviously, in this brief summary i have to be far from being complete. Let p1,p2 and q be measures on the same measurable set x. Whend1, bothrdandrvequal twice the area between the usual lorenz curve and the line of zero disparity.
Majorization, schurconvex functions, graphs, second zagreb index. Minimizing churn in distributed systems acm sigcomm. Indeed, many majorization concepts have been reinvented and. Introduction prior to the appearance of the celebrated volume inequalities. A new characterization ofrdas proportional to a univariate gini index for a particular linear combination of attributes relates it to the lorenz zonoid. The lorenz curve in economics and econometrics semantic. Americas secret establishment an introduction to the order of skull and bones by antony c. It all goes back to a brief paper published by max lorenz in the publication of the american. Introduction in last years, a lot of papers was dedicated to majorization theory, that was scattered in journals in a wide variety of. Arnold and villasefior 1989 have studied some lorenzorder relations among order statistics from the same parent powerfunction distribution. The pre sent set of notes is designed for a one quarter course introducing majorization and the lorenz order. The exposition of multivariate lorenz orderings sets it apart from existing treatments.
Revising and expanding on an earlier monograph, majorization and the lorenz order. The inequality principles of dalton, especially the transfer or robin hood principle, are given. We write p2 q p1 if the lorenz diagram of p1,q is a subset of the lorenz diagram of p2,q. A bivariate pareto distribution with freund s dependence. The lorenz order in the space of distribution functions. There are several ways to characterize the ordering of majorization. My interest in majorization was first spurred by ingram aikins proclivity for finding schur convex functions lurking in the problem section of. The pre sent set of notes is designed for a one quarter. This greatly expanded new edition includes recent research on stochastic, multivariate and group majorization, lorenz order, and applications in physics and chemistry, in economics and political science, in matrix inequalities, and in probability and statistics. It shows students how to analyze global political events using theoretical approaches both mainstream and alternative and emphasizes non state actors more than other textbooks. The text of this book was adapted from usa history in brief, gps catalog no. Arnold, 9780387965925, available at book depository with free delivery worldwide. Theory of majorization and its applications marshall and olkin,1979many researchers were unaware of the rich bodyof literature related to majorization that was scattered in journals in a wide variety of.
For example, a wealth distribution is lorenz greater than another iff its lorenz curve lies below the other. The dotdashed curve is the lorenz curve of the most aggregated distribution compared with this negative binomial distribution having the. Arnold and villasefior 1989 have studied some lorenz order relations among order statistics from the same parent powerfunction distribution. The dotted line is a lorenz curve of a distribution supported on two values.
Hardylittlewoodpolyas inequality and a new concept of. We show that the systems temperature is significantly suppressed due to both the entropy majorization theorem in quantum information science and the entropy conservation law in reversible adiabatic processes. The inequality principles of dalton, especially the transfer or robin hood principle, are given appropriate prominence. In this section we give a short account of transfer principles and majoriza tion, and of their. Order theory mark dean lecture notes for fall 2015 phd class in decision theory brown university 1 introduction we are going to begin the class with some mathematical preliminaries. Entropy majorization, thermal adiabatic theorem, and. Majorization in economic disparity measures sciencedirect. This paper generalizes their work by obtaining conditions for the ordering of two order statistics from different powerfunction distributions with varying sample sizes.
On truncated versions of certain measures of inequality and. Indeed, many majorization concepts have been reinvented and used in di. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Majorization under constraints and bounds on the second. Pdf majorization and dynamics of continuous distributions. I have found the income distribution context to be quite convenient for discussion of inequality orderings.
Multivariate convex orderings, dependence, and stochastic. My interest in majorization was first spurred by ingram aikins proclivity for finding schur convex functions lurking in the problem section of every issue of the american mathematical monthly. Chapter 1 a brief introduction here we meet string theory for the. These include characterizations of income distributions in terms of families of inequality measures, lorenz ordering of multiparameter distributions in terms of their parameters, probability inequalities for. The approach to the analysis of income inequality based on majorization which dates back to lorenz 26 has been used by atkinson 4 and dasgupta, sen and starrett 9 where conditions equivalent to the lorenz majorization criterion were obtained. On truncated versions of certain measures of inequality. Sordo references b c arnold 1987 majorization and the lorenz order. We take the onedimensional transversefield ising model and the spinless. We extend one of the main findings in bossmann et al. Reflection groups, generalized schur functions, and the geometry of majorization eaton, morris l. The lorenz diagram of p1,q is a subset of the lorenz diagram of p2,q if and only if there exists a markov operator that transforms p2 into p1 and leaves q invariant. L vector with values of the ordinary lorenz curve l.
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