# A unified Lorenz-type approach to divergence and dependence

• Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), 1994

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## Abstract

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AbstractThe paper deals with function-valued and numerical measures of absolute and directed divergence of one probability measure from another. In case of absolute divergence, some new results are added to the known ones to form a unified structure. In case of directed divergence, new concepts are introduced and investigated. It is shown that the notions of absolute and directed divergences complement each other and provide a good insight into the extent and the type of discrepancy between two distributions. Consequently, these measures applied together to suitably chosen pairs of distributions prove useful to express such statistical concepts as inequality, dependence, and departures from proportionality.CONTENTS   Introduction.......................................................................................................................................51. Divergence of probability measures................................................................................................8   1.1. Divergence of probability measures connected with two-class classification problems...............8   1.2. Concentration curve and its link with the Neyman-Pearson curve.............................................10   1.3. Divergence ordering ${⪯}_{NP}$.................................................................................................112. Link between divergence and inequality..........................................................................................13   2.1. Initial inequality axioms..............................................................................................................13   2.2. The Lorenz curve for nonnegative random variables..................................................................14   2.3. Inequality ordering ${⪯}_{L}$......................................................................................................15   2.4. Inequality versus divergence......................................................................................................17   2.5. Ratio variables...........................................................................................................................193. Link between divergence and dependence.....................................................................................20   3.1. Preliminary remarks...................................................................................................................20   3.2. Dependence ordering ${⪯}_{D}$................................................................................................22   3.3. Orderings related to ${⪯}_{D}$...................................................................................................224. Link between divergence and proportional representation.............................................................24   4.1. Formulation of the problem and definition of the ordering ${⪯}_{x}$..........................................24   4.2. Minimal elements for ${⪯}_{x}$...................................................................................................26   4.3. Maximal elements for ${⪯}_{x}$..................................................................................................295. Directed concentration of probability measures.............................................................................30   5.1. Directed concentration curve....................................................................................................30   5.2. Grade transformation of a random variable...............................................................................34   5.3. Correlation and ratio curves......................................................................................................35   5.4. Directed departure from proportionality....................................................................................406. Numerical measures relating to divergence....................................................................................42   6.1. Numerical inequality measures..................................................................................................42   6.2. Numerical measures of divergence............................................................................................44   6.3. Numerical measures of directed divergence..............................................................................45   6.4. Numerical measures of dependence..........................................................................................47   6.5. Numerical measures of departures from proportional representation.......................................49   References........................................................................................................................................51   Index of symbols................................................................................................................................541991 Mathematics Subject Classification: 62H30, 62H20, 90A19.

## How to cite

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Teresa Kowalczyk. A unified Lorenz-type approach to divergence and dependence. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1994. <http://eudml.org/doc/268506>.

@book{TeresaKowalczyk1994,
abstract = {AbstractThe paper deals with function-valued and numerical measures of absolute and directed divergence of one probability measure from another. In case of absolute divergence, some new results are added to the known ones to form a unified structure. In case of directed divergence, new concepts are introduced and investigated. It is shown that the notions of absolute and directed divergences complement each other and provide a good insight into the extent and the type of discrepancy between two distributions. Consequently, these measures applied together to suitably chosen pairs of distributions prove useful to express such statistical concepts as inequality, dependence, and departures from proportionality.CONTENTS   Introduction.......................................................................................................................................51. Divergence of probability measures................................................................................................8   1.1. Divergence of probability measures connected with two-class classification problems...............8   1.2. Concentration curve and its link with the Neyman-Pearson curve.............................................10   1.3. Divergence ordering $⪯_\{NP\}$.................................................................................................112. Link between divergence and inequality..........................................................................................13   2.1. Initial inequality axioms..............................................................................................................13   2.2. The Lorenz curve for nonnegative random variables..................................................................14   2.3. Inequality ordering $⪯_\{L\}$......................................................................................................15   2.4. Inequality versus divergence......................................................................................................17   2.5. Ratio variables...........................................................................................................................193. Link between divergence and dependence.....................................................................................20   3.1. Preliminary remarks...................................................................................................................20   3.2. Dependence ordering $⪯_\{D\}$................................................................................................22   3.3. Orderings related to $⪯_\{D\}$...................................................................................................224. Link between divergence and proportional representation.............................................................24   4.1. Formulation of the problem and definition of the ordering $⪯_\{x\}$..........................................24   4.2. Minimal elements for $⪯_\{x\}$...................................................................................................26   4.3. Maximal elements for $⪯_\{x\}$..................................................................................................295. Directed concentration of probability measures.............................................................................30   5.1. Directed concentration curve....................................................................................................30   5.2. Grade transformation of a random variable...............................................................................34   5.3. Correlation and ratio curves......................................................................................................35   5.4. Directed departure from proportionality....................................................................................406. Numerical measures relating to divergence....................................................................................42   6.1. Numerical inequality measures..................................................................................................42   6.2. Numerical measures of divergence............................................................................................44   6.3. Numerical measures of directed divergence..............................................................................45   6.4. Numerical measures of dependence..........................................................................................47   6.5. Numerical measures of departures from proportional representation.......................................49   References........................................................................................................................................51   Index of symbols................................................................................................................................541991 Mathematics Subject Classification: 62H30, 62H20, 90A19.},
author = {Teresa Kowalczyk},
keywords = {function-valued measures of divergence; numerical measures of divergence; concentration curves; Neyman-Pearson curve; Lorenz curve; orderings; absolute divergence; directed divergence; inequality; dependence; departures from proportionality},
language = {eng},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {A unified Lorenz-type approach to divergence and dependence},
url = {http://eudml.org/doc/268506},
year = {1994},
}

TY - BOOK
AU - Teresa Kowalczyk
TI - A unified Lorenz-type approach to divergence and dependence
PY - 1994
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - AbstractThe paper deals with function-valued and numerical measures of absolute and directed divergence of one probability measure from another. In case of absolute divergence, some new results are added to the known ones to form a unified structure. In case of directed divergence, new concepts are introduced and investigated. It is shown that the notions of absolute and directed divergences complement each other and provide a good insight into the extent and the type of discrepancy between two distributions. Consequently, these measures applied together to suitably chosen pairs of distributions prove useful to express such statistical concepts as inequality, dependence, and departures from proportionality.CONTENTS   Introduction.......................................................................................................................................51. Divergence of probability measures................................................................................................8   1.1. Divergence of probability measures connected with two-class classification problems...............8   1.2. Concentration curve and its link with the Neyman-Pearson curve.............................................10   1.3. Divergence ordering $⪯_{NP}$.................................................................................................112. Link between divergence and inequality..........................................................................................13   2.1. Initial inequality axioms..............................................................................................................13   2.2. The Lorenz curve for nonnegative random variables..................................................................14   2.3. Inequality ordering $⪯_{L}$......................................................................................................15   2.4. Inequality versus divergence......................................................................................................17   2.5. Ratio variables...........................................................................................................................193. Link between divergence and dependence.....................................................................................20   3.1. Preliminary remarks...................................................................................................................20   3.2. Dependence ordering $⪯_{D}$................................................................................................22   3.3. Orderings related to $⪯_{D}$...................................................................................................224. Link between divergence and proportional representation.............................................................24   4.1. Formulation of the problem and definition of the ordering $⪯_{x}$..........................................24   4.2. Minimal elements for $⪯_{x}$...................................................................................................26   4.3. Maximal elements for $⪯_{x}$..................................................................................................295. Directed concentration of probability measures.............................................................................30   5.1. Directed concentration curve....................................................................................................30   5.2. Grade transformation of a random variable...............................................................................34   5.3. Correlation and ratio curves......................................................................................................35   5.4. Directed departure from proportionality....................................................................................406. Numerical measures relating to divergence....................................................................................42   6.1. Numerical inequality measures..................................................................................................42   6.2. Numerical measures of divergence............................................................................................44   6.3. Numerical measures of directed divergence..............................................................................45   6.4. Numerical measures of dependence..........................................................................................47   6.5. Numerical measures of departures from proportional representation.......................................49   References........................................................................................................................................51   Index of symbols................................................................................................................................541991 Mathematics Subject Classification: 62H30, 62H20, 90A19.
LA - eng
KW - function-valued measures of divergence; numerical measures of divergence; concentration curves; Neyman-Pearson curve; Lorenz curve; orderings; absolute divergence; directed divergence; inequality; dependence; departures from proportionality
UR - http://eudml.org/doc/268506
ER -

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