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Tytuł książki

## A unified Lorenz-type approach to divergence and dependence

### Seria

Rozprawy Matematyczne tom/nr w serii: 335 wydano: 1994

### Abstrakty

EN

CONTENTS
Introduction.......................................................................................................................................5
1. 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 $⪯_{NP}$.................................................................................................11
2. 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...........................................................................................................................19
3. Link between divergence and dependence.....................................................................................20
3.1. Preliminary remarks...................................................................................................................20
3.2. Dependence ordering $⪯_{D}$................................................................................................22
3.3. Orderings related to $⪯_{D}$...................................................................................................22
4. 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 $⪯_{x}$..................................................................................................29
5. 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....................................................................................40
6. 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

### Tematy

Kategoryzacja MSC:

Warszawa

### Seria

Rozprawy Matematyczne tom/nr w serii: 335

54

### Opis fizyczny

Dissertationes Mathematicae, Tom CCCXXXV

wydano
1994
otrzymano
1993-07-20
poprawiono
1994-01-28

### Twórcy

autor
• Institute of Computer Science, Polish Academy of Sciences, P.O. Box 22, J. Ordona 21, 01-237 Warszawa, Poland

### Bibliografia

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• S. M. Ali and S. D. Silvey (1966), A general class of coefficients of divergence of one distribution from another, ibid., 28, 131-142.
• B. C. Arnold (1987), Majorization and the Lorenz Order: a Brief Introduction, Lecture Notes in Statist. 43, Dekker.
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• R. C. Blitz and J. A. Brittain (1964), An extension of the Lorenz diagram to the correlation of two variables, Metron 23 (1964), 137-143.
• H. Block, A. Sampson and T. Savits (eds.) (1990), Topics in Statistical Dependence, IMS Lecture Notes Monograph Ser., Inst. Math. Statist., Hayward.
• Z. Bondarczuk, T. Kowalczyk, E. Pleszczyńska and W. Szczesny (1994), Evaluating departures from fair representation, Appl. Stochastic Models Data Anal., to appear.
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• T. Bromek and T. Kowalczyk (1990), A decision approach to ordering stochastic dependence, in: A. Sampson (ed.), Topics in Statistical Dependence, IMS Lecture Notes Monograph Ser., Inst. Math. Statist., Hayward, 103-109.
• M. Chandra and N. D. Singpurwalla (1981), Relationship between some notions which are common to reliability and economics, Math. Oper. Res. 6, 113-121.
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• A. Ciok, T. Kowalczyk, E. Pleszczyńska and W. Szczesny (1994), Inequality measures in data analysis, Archiwum Informatyki Teoretycznej i Stosowanej, to appear.
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• J. E. Foster (1985), Inequality measurement, in: Proc. Sympos. Appl. Math. 33, 31-68.
• V. Gafrikova and T. Kowalczyk (1994), Links between measuring divergence and inequality, Metron, to appear.
• D. M. Grove (1980), A test of independence against a class of ordered alternatives in a 2 × C contingency table, J. Amer. Statist. Assoc. 75, 454-459.
• H. Joe (1985), An ordering of dependence for contingency tables, Linear Algebra Appl. 70, 89-103.
• H. Joe (1987), Majorization, randomness and dependence for multivariate distribution, Ann. Probab. 15, 1217-1225.
• H. Joe (1990), Majorization and divergence, J. Math. Anal. Appl. 148, 287-305.
• B. Klefsjö (1984), Reliability interpretations of some concepts from economics, Naval Res. Logist. 31, 301-308.
• T. Kowalczyk (1977), General definition and sample counterparts of monotonic dependence functions of bivariate distributions, Math. Oper. Statist. Ser. Statist. 8, 351-365.
• T. Kowalczyk (1990), On measuring heterogeneity in m × k contingency tables, in: Proc. DIANA III, Conference of Discriminant Analysis and Other Methods of Data Classification, Bechyne, 111-121.
• T. Kowalczyk and J. Mielniczuk (1990), Neyman-Pearson curves, properties and estimation, preprint 683, IPI PAN.
• T. Kowalczyk and E. Pleszczyńska (1977), Monotonic dependence functions of bivariate distributions, Ann. Statist. 5, 1221-1227.
• T. Kowalczyk, E. Pleszczyńska and W. Szczesny (1991), Evaluation of stochastic dependence, in: Statistical Inference: Theory and Practice, Theory Decis. Lib. Ser. B: Math. Statist. Methods 17, Reidel, 106-136.
• E. L. Lehmann (1959), Testing Statistical Hypotheses, Wiley, New York.
• E. L. Lehmann (1966), Some concepts of dependence, Ann. Math. Statist. 37, 1137-1153.
• R. Lerman and S. Yitzaki (1984), A note on the calculation and interpretation of the Gini index, Econom. Lett. 15, 363-368.
• C. E. Rao (1982), Diversity and dissimilarity coefficients: a unified approach, Theoret. Population Biol. 21, 24-43.
• A. Raveh (1989), Gini correlation as a measure of monotonicity and two of its usages, Comm. Statist. Theory Methods 18 (4), 1415-1423.
• E. Regazzini (1990), Concentration comparisons between probability measures, Instituto per le Applicazioni della Matematica e dell'Informatica, preprint 90.15, Milano.
• M. Scarsini (1990), An ordering of dependence, in: A. Sampson (ed.), Topics in Statistical Dependence, IMS Lecture Notes Monograph Ser., Inst. Math. Statist., Hayward, 403-414.
• E. Schechtman and S. Yitzaki (1987), A measure of association based on Gini's mean difference, Comm. Statist. Theory Methods 16 (1), 207-231.
• W. Szczesny (1991), On the performance of a discriminant function, J. Classification 8, 201-215.
• T. Taguchi (1987), On the structure of multivariate concentration - some relationships among the concentration surface and two variate mean difference and regressions, Comput. Statist. Data Anal. 6, 307-334.
• N. White (1986), Segregation and diversity measures in population distribution, Population Index 52, 198-221.

 EN

### Uwagi

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

bwmeta1.element.zamlynska-3f9b39c2-f5b5-4621-ab6d-71696fcd7fb6

ISSN
0012-3862

### Kolekcja

DML-PL
Zawartość książki

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