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E-raamat: Pareto Distributions

(University of California, Riverside, USA)
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Since the publication of the first edition over 30 years ago, the literature related to Pareto distributions has flourished to encompass computer-based inference methods.Pareto Distributions, Second Edition provides broad, up-to-date coverage of the Pareto model and its extensions. This edition expands several chapters to accommodate recent results and reflect the increased use of more computer-intensive inference procedures.

New to the Second Edition

  • New material on multivariate inequality
  • Recent ways of handling the problems of inference for Pareto models and their generalizations and extensions
  • New discussions of bivariate and multivariate income and survival models

This book continues to provide researchers with a useful resource for understanding the statistical aspects of Pareto and Pareto-like distributions. It covers income models and properties of Pareto distributions, measures of inequality for studying income distributions, inference procedures for Pareto distributions, and various multivariate Pareto distributions existing in the literature.

List of Figures xvii
List of Tables xix
Preface to the First Edition xxi
Preface to the Second Edition xxiii
1 Historical sketch with emphasis on income modeling 1(18)
1.1 Introduction
1(1)
1.2 The First Steps
2(5)
1.3 The Modern Era
7(12)
2 Models for income distributions 19(22)
2.1 What Is a Model?
19(1)
2.2 The Law of Proportional Effect (Gibrat)
20(1)
2.3 A Markov Chain Model (Champernowne)
21(3)
2.4 The Coin Shower (Ericson)
24(1)
2.5 An Open Population Model (Rutherford)
25(2)
2.6 The Yule Distribution (Simon)
27(3)
2.7 Income Determined by Inherited Wealth (Wold-Whittle)
30(1)
2.8 The Pyramid (Lydall)
30(1)
2.9 Competitive Bidding for Employment (Arnold and Laguna)
31(2)
2.10 Other Models
33(2)
2.11 Parametric Families for Fitting Income Distributions
35(6)
3 Pareto and related heavy-tailed distributions 41(76)
3.1 Introduction
41(1)
3.2 The Generalized Pareto Distributions
41(6)
3.3 Distributional Properties
47(8)
3.3.1 Modes
47(1)
3.3.2 Moments
47(2)
3.3.3 Transforms
49(1)
3.3.4 Standard Pareto Distribution
50(1)
3.3.5 Infinite Divisibility
50(1)
3.3.6 Reliability, P(X1 <X2)
50(1)
3.3.7 Convolutions
51(2)
3.3.8 Products of Pareto Variables
53(1)
3.3.9 Mixtures, Random Sums and Random Extrema
54(1)
3.4 Order Statistics
55(9)
3.4.1 Ratios of Order Statistics
57(2)
3.4.2 Moments
59(4)
3.4.3 Moments in the Presence of Truncation
63(1)
3.5 Record Values
64(3)
3.6 Generalized Order Statistics
67(2)
3.7 Residual Life
69(2)
3.8 Asymptotic Results
71(7)
3.8.1 Order Statistics
71(2)
3.8.2 Convolutions
73(1)
3.8.3 Record Values
74(1)
3.8.4 Generalized Order Statistics
74(1)
3.8.5 Residual Life
75(1)
3.8.6 Geometric Minimization and Maximization
75(2)
3.8.7 Record Values Once More
77(1)
3.9 Characterizations
78(22)
3.9.1 Mean Residual Life
78(4)
3.9.2 Truncation Equivalent to Resealing
82(1)
3.9.3 Inequality Measures
83(1)
3.9.4 Under-reported Income
84(2)
3.9.5 Functions of Order Statistics
86(7)
3.9.6 Record Values
93(1)
3.9.7 Generalized Order Statistics
94(3)
3.9.8 Entropy Maximization
97(1)
3.9.9 Pareto (III) Characterizations
98(1)
3.9.10 Two More Characterizations
99(1)
3.10 Related Distributions
100(10)
3.11 The Discrete Pareto (Zipf) Distribution
110(5)
3.11.1 Zeta Distribution
110(1)
3.11.2 Zipf Distributions
111(1)
3.11.3 Simon Distributions
112(2)
3.11.4 Characterizations
114(1)
3.12 Remarks
115(2)
4 Measures of inequality 117(106)
4.1 Apologia for Prolixity
117(1)
4.2 Common Measures of Inequality of Distributions
118(52)
4.2.1 The Lorenz Curve
121(2)
4.2.2 Inequality Measures Derived from the Lorenz Curve
123(12)
4.2.3 The Effect of Grouping
135(4)
4.2.4 Multivariate Lorenz Curves
139(6)
4.2.5 Moment Distributions
145(3)
4.2.6 Related Reliability Concepts
148(1)
4.2.7 Relations Between Inequality Measures
149(1)
4.2.8 Inequality Measures for Specific Distributions
149(8)
4.2.9 Families of Lorenz Curves
157(9)
4.2.10 Some Alternative Inequality Curves
166(4)
4.3 Inequality Statistics
170(24)
4.3.1 Graphical Techniques
171(4)
4.3.2 Analytic Measures of Inequality
175(8)
4.3.3 The Sample Gini Index
183(2)
4.3.4 Sample Lorenz Curve
185(4)
4.3.5 Further Sample Measures of Inequality
189(4)
4.3.6 Relations Between Sample Inequality Measures
193(1)
4.4 Inequality Principles and Utility
194(27)
4.4.1 Inequality Principles
195(1)
4.4.2 Transfers, Majorization and the Lorenz Order
196(7)
4.4.3 How Transformations Affect Inequality
203(6)
4.4.4 Weighting and Mixing
209(2)
4.4.5 Lorenz Order within Parametric Families
211(1)
4.4.6 The Lorenz Order and Order Statistics
212(2)
4.4.7 Related Orderings
214(2)
4.4.8 Multivariate Extensions of the Lorenz Order
216(5)
4.5 Optimal Income Distributions
221(2)
5 Inference for Pareto distributions 223(76)
5.1 Introduction
223(1)
5.2 Parameter Estimation
224(40)
5.2.1 Maximum Likelihood
224(3)
5.2.2 Best Unbiased and Related Estimates
227(6)
5.2.3 Moment and Quantile Estimates
233(3)
5.2.4 A Graphical Technique
236(1)
5.2.5 Bayes Estimates
236(7)
5.2.6 Bayes Estimates Based on Other Data Configurations
243(3)
5.2.7 Bayes Prediction
246(2)
5.2.8 Empirical Bayes Estimation
248(1)
5.2.9 Miscellaneous Bayesian Contributions
249(1)
5.2.10 Maximum Likelihood for Generalized Pareto Distributions
249(5)
5.2.11 Estimates Using the Method of Moments and Estimating Equations for Generalized Pareto Distributions
254(5)
5.2.12 Order Statistic Estimates for Generalized Pareto Distributions
259(4)
5.2.13 Bayes Estimates for Generalized Pareto Distributions
263(1)
5.3 Interval Estimates
264(4)
5.4 Parametric Hypotheses
268(1)
5.5 Tests to Aid in Model Selection
269(6)
5.6 Specialized Techniques for Various Data Configurations
275(8)
5.7 Grouped Data
283(5)
5.8 Inference for Related Distributions
288(11)
5.8.1 Zeta Distribution
289(1)
5.8.2 Simon Distributions
289(3)
5.8.3 Waring Distribution
292(1)
5.8.4 Under-reported Income Distributions
293(3)
5.8.5 Inference for Flexible Extensions of Pareto Models
296(2)
5.8.6 Back to Pareto
298(1)
6 Multivariate Pareto distributions 299(54)
6.0 Introduction
299(1)
6.1 A Hierarchy of Multivariate Pareto Models
299(11)
6.1.1 Mardia's First Multivariate Pareto Model
299(1)
6.1.2 A Hierarchy of Generalizations
300(4)
6.1.3 Distributional Properties of the Generalized Multivariate Pareto Models
304(5)
6.1.4 Some Characterizations of Multivariate Pareto Models
309(1)
6.2 Alternative Multivariate Pareto Distributions
310(18)
6.2.1 Mixtures of Weibull Variables
310(2)
6.2.2 Transformed Exponential Variables
312(1)
6.2.3 Trivariate Reduction
313(2)
6.2.4 Geometric Minimization and Maximization
315(4)
6.2.5 Building Multivariate Pareto Models Using Independent Gamma Distributed Components
319(2)
6.2.6 Other Bivariate and Multivariate Pareto Models
321(2)
6.2.7 General Classes of Bivariate Pareto Distributions
323(2)
6.2.8 A Flexible Multivariate Pareto Model
325(2)
6.2.9 Matrix-variate Pareto Distributions
327(1)
6.3 Related Multivariate Models
328(8)
6.3.1 Conditionally Specified Models
328(4)
6.3.2 Multivariate Hidden Truncation Models
332(2)
6.3.3 Beta Extensions
334(1)
6.3.4 Kumaraswamy Extensions
334(1)
6.3.5 Multivariate Semi-Pareto Distributions
335(1)
6.4 Pareto and Semi-Pareto Processes
336(4)
6.5 Inference for Multivariate Pareto Distributions
340(10)
6.5.1 Estimation for Mardia's Multivariate Pareto Families
340(2)
6.5.2 Estimation for More General Multivariate Pareto Families
342(5)
6.5.3 A Confidence Interval Based on a Multivariate Pareto Sample
347(2)
6.5.4 Remarks
349(1)
6.6 Multivariate Discrete Pareto (Zipf) Distributions
350(3)
A Historical income data sources 353(6)
B Two representative data sets 359(10)
C A quarterly household income data set 369(2)
References 371(38)
Subject Index 409(12)
Author Index 421
Barry C. Arnold is a distinguished professor in the Department of Statistics at the University of California, Riverside. Dr. Arnold is a fellow of the American Statistical Association and the Institute of Mathematical Statistics and an elected member of the International Statistical Institute.
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