Correlation and Dependence

, by Mari, Dominique Drouet; Kotz, Samuel

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ISBN: 9781860942648 | 1860942644
Cover: Hardcover
Copyright: 6/1/2001

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(Imperial College Press) A graduate text on correlation and dependence concepts and measures, designed to help remedy the lack of such texts for students in statistics, engineering, and mathematics. Takes a tour of this neglected subject, requiring some background in mathematical statistics and integral calculus. DLC: Correlation (Statistics).

Preface

vii

Notations and Definitions

(6)

Notations

(2)

Definitions

(4)

Correlation and Dependence: An Introspection

(24)

Independence

(3)

Zero Correlation Versus Dependence

(8)

Linear relationship

(2)

Non-linear relationship

(3)

A technical discussion

(2)

Some Geometrical Examples

(4)

Some Further Historical Remarks

(3)

A Brief Tour of Early Applications and Misinterpretations

(6)

Concepts of Dependence and Stochastic Ordering

(34)

Introduction

(2)

Concepts of Positive Dependence

(10)

The Kimeldorf and Sampson conditions

(1)

The positive quadrant dependence (PQD)

(1)

Positive upper or lower orthant dependence

(1)

Positive upper or lower set dependence

(1)

Association

(1)

Positive function dependence

(1)

Positive regression dependence (PRD)

(1)

Multivariate case

(1)

The Lihelihood ratio dependence (LRD)

(1)

Dependences DTP (m,n)

(1)

Positive dependence by mixture

(1)

Implications of the concepts

(1)

Lower and upper tail dependence

(1)

Negative Dependence for More than Two Variables

(7)

NUOD and NLOD

(1)

Definition from RR2

(1)

Structural condition

(2)

Negative association

(2)

Negatively superadditive dependence

(2)

Setwise Dependence

(3)

Setwise upper orthant and setwise upper set positive dependences

(1)

Setwise association

(1)

Setwise dependence by mixture

(1)

Extension to the setwise negative dependence

(1)

Other Approaches

(1)

Positive Dependence Orderings

(9)

Ordering based on PQD

(1)

Conditions on ordering

(1)

Ordering defined by PRD

(1)

Association ordering

(1)

PDD-ordering

(1)

Orderings defined from DTP (0,1) and LRD

(2)

Integral stochastic orderings

(1)

Supermodular ordering

(1)

Directionally convex ordering

(1)

Generating a family of partial orderings

(2)

Bayesian Approach to Stochastic Dependence

(2)

Copulas

(48)

Introduction

(1)

Definition and Some Properties

(2)

The Frechet Bounds

(2)

Lower and upper Frechet bounds in the family F(F1, F2) of bivariate distributions with common marginals

(1)

The Frechet bounds for a copula

(1)

Examples

(3)

Construction of a Copula

(11)

The Ruschendorf method

(1)

Application to polynomial couplas

(1)

Approximation of a copula by a polynomial copula

(1)

Other examples

(2)

Models defined from a distortion function

(1)

Frailty models

(2)

Marshall and Olkin's generalization

(4)

Archimedean Copulas

(10)

Definition and basic properties

(1)

Examples

(2)

A characterization of Archimedean copulas

(1)

The limit of a sequence of Archimedean copulas

(1)

Characterization of Archimedean copulas by their diagonal copulas

(2)

Fitting an observed distribution with an Archimedean copula

(1)

Characterization of an Archimedean copula by the cumulative distribution function of Z = C(U, V)

(2)

Archimedean copulas with two parameters

(1)

Archimax Copulas

(2)

Extreme value distribution and extreme value copula

(1)

Definition of Archimax copulas

(1)

Construction of bivariate distributions belonging to a predetermined domain of attraction

(1)

Examples

(1)

Copulas with Discontinuity Constraints

(5)

Piecewise additive copulas

(1)

Piecewise quadratic copulas

(1)

Quadratic copulas with holes

(1)

Admissible rectangles

(1)

The squeeze algorithm

100

(1)

Copulas with More than Two Variables

101

(9)

m-dimensional Archimedean copulas

102

(1)

An application

102

(3)

Generation of a 3-dimensional copula from its 2-dimensional marginals

105

(1)

Compatibility of marginals

105

(1)

Truncation invariance

105

(3)

Linkages

108

(2)

Simulation Procedures

110

(3)

The general case

110

(1)

Archimedean copulas

110

(1)

Archimax distributions

111

(1)

Marshall and Olkin's mixture of distributions

112

(1)

Three-dimensional copulas with truncation invariance

112

(1)

Farlie-Gumbel-Morgenstern Models of Dependence

113

(36)

Introduction

113

(1)

Initial Definition

114

(1)

Regression and Correlation

115

(2)

Iterations

117

(2)

Dependence Properties

119

(1)

A Class of n-variate FGM Distributions

119

(11)

A class of bivariate FGM distributions with Weibull marginal distributions

122

(3)

A class of three-variate distributions with Weibull marginal distributions

125

(4)

FGM n-variate distributions with Weibull marginals

129

(1)

Further Extensions

130

(11)

Huang and Kotz Extensions

131

(1)

Sarmanov's extension

132

(2)

Sarmanov-Lee extension

134

(1)

Bairamov-Kotz extensions

135

(1)

Lai and Xie extension

136

(1)

Bairamov-Kotz-Bekci generalization

137

(2)

Concomitants of order statistics

139

(2)

FGM Sequences

141

(8)

Global Versus Local Dependence between Random Variables

149

(54)

Introduction

149

(1)

Global Measures of Dependence

150

(21)

Desirable properties of a measure of dependence

150

(1)

Covariance, Q-covariance

151

(2)

The coefficient of linear correlation ρ

153

(1)

The case when (X, Y) is bivariate normal

153

(2)

Correlation and extremal properties of normal distributions

155

(1)

ρ and the moment of inertia around the line D1: {y = x}

155

(1)

A geometric interpretation

156

(1)

ρ and concepts of dependence

157

(1)

The ρs of Spearman and its connection with the PQD concept

157

(1)

A geometric interpretation of ρs

158

(1)

Estimation of ρs

158

(1)

Schweizer-Wolff's index of dependence

159

(1)

The Kendall τ and its connection with LRD property

159

(1)

Estimation of τ

160

(1)

The Blomqvist medial coefficient

161

(1)

τ ρs, β and ordering on the distributions

161

(1)

Constructing other global measures

162

(1)

Indices for more than two variables

162

(1)

Mutual information, relative entropy and derivatives measures

163

(1)

Definitions

163

(2)

Examples

165

(2)

Lin's measure of association

167

(1)

Zografos's measure of association

168

(3)

Local Indices of Dependence

171

(19)

Motivation

171

(1)

Local definition of the dependence

171

(1)

Local ρs and τ

172

(1)

Local correlation coefficient of Bjerve and Doksum

172

(1)

Motivation and historical remarks

172

(1)

Definition, properties and limits

173

(1)

Estimations and properties of the estimators

174

(1)

Correlation ratio

175

(1)

Local dependence function of Bairamov and Kotz

175

(2)

Measures of the tail dependence

177

(1)

Several local indices applicable in survival analysis

178

(1)

The covariance function of Prentice and Cai

178

(1)

The conditional covariance rate of Dabrowska

179

(4)

Θ : the ratio of two conditional hazards

183

(5)

Other measures derived from Θ

188

(1)

A Local measure of LRD dependence γf

189

(1)

Non-parametric Estimation of Local Indices

190

(4)

The univariate case: estimation of H(x)

190

(1)

Bivariate case and conditional hazards

191

(1)

Consistency and asymptotic normality of the estimate of h(x, y)

192

(1)

Consistency and asymptotic normality of the two conditional hazards

193

(1)

Estimation of the indices l and Θ

194

(1)

A Search for the Localisation of the Maximal Association

194

(9)

Lower and upper tail dependence for the three distributions

196

(1)

Θ and the remaining dependence

197

(2)

Index γ and the instantaneous dependence

199

(1)

Simulation of the survival bivariate distributions and estimation of Θ

200

(3)

Bibliography

203

(14)

Index

217

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Correlation and Dependence

Summary

Table of Contents

Supplemental Materials