A unified Lorenz-type approach to divergence and dependence

Teresa Kowalczyk

GDML_Books, (1994), p.

Résumé

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................................................................................................................................54

1991 Mathematics Subject Classification: 62H30, 62H20, 90A19.

Détail

Zbl 0855.62045

EUDML-ID : urn:eudml:doc:268506