Correlation and dependence:
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
London
Imperial College Press
2001
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009555886&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XIV, 219 S. graph. Darst. |
| ISBN: | 1860942644 |
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| 100 | 1 | |a Drouet Mari, Dominique |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Correlation and dependence |c Dominique Drouet Mari ; Samuel Kotz |
| 264 | 1 | |a London |b Imperial College Press |c 2001 | |
| 300 | |a XIV, 219 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 7 | |a Correlatieanalyse |2 gtt | |
| 650 | 4 | |a Corrélation (Statistique) | |
| 650 | 4 | |a Dépendance (Statistique) | |
| 650 | 7 | |a Inferência estatística |2 larpcal | |
| 650 | 4 | |a Correlation (Statistics) | |
| 650 | 4 | |a Dependence (Statistics) | |
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| 700 | 1 | |a Kotz, Samuel |d 1930-2010 |e Verfasser |0 (DE-588)119529653 |4 aut | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-009555886 | |
Datensatz im Suchindex
| DE-BY-TUM_call_number | 0048 MAT 628f 2005 A 4122 |
|---|---|
| DE-BY-TUM_katkey | 1750103 |
| DE-BY-TUM_location | LSB |
| DE-BY-TUM_media_number | 040010201900 |
| _version_ | 1821933268817149952 |
| adam_text | Contents
Preface vii
Chapter 1 Notations and Definitions 1
LI Notations..................................................... 1
1.2 Definitions .................................................. 3
Chapter 2 Correlation and Dependence : An Introspection 7
2.1 Independence ................................................. 7
2.2 Zero Correlation Versus Dependence........................... 10
2.2.1 Linear relationship................................... 11
2.2.2 Non-linear relationship............................... 13
2.2.3 A technical discussion................................ 16
2.3 Some Geometrical Examples ................................... 18
2.4 Some Further Historical Remarks.............................. 22
2.5 A Brief Tour of Early Applications and Misinterpretations . . 25
Chapter 3 Concepts of Dependence and Stochastic Ordering 31
3.1 Introduction ................................................ 31
3.2 Concepts of Positive Dependence.............................. 33
3.2.1 The Kimeldorf and Sampson conditions.................. 33
3.2.2 The positive quadrant dependence (PQD) ............... 34
3.2.2.1 Positive upper or lower orthant dependence . 35
3.2.2.2 Positive upper or lower set dependence .... 36
3.2.3 Association........................................... 36
3.2.4 Positive function dependence.......................... 37
ix
X
Contents
3.2.5 Positive regression dependence (PRD).................. 38
3.2.5.1 Multivariate case............................ 39
3.2.6 The Lihelihood ratio dependence (LRD) ................ 39
3.2.7 Dependences DTP(m,n).................................. 40
3.2.8 Positive dependence by mixture ....................... 41
3.2.9 Implications of the concepts.......................... 42
3.2.10 Lower and upper tail dependence ...................... 43
3.3 Negative Dependence for More than Two Variables ............. 43
3.3.1 NUOD and NLOD ........................................ 44
3.3.2 Definition from RR2................................... 44
3.3.3 Structural condition.................................. 44
3.3.4 Negative association.................................. 46
3.3.5 Negatively superadditive dependence .................. 48
3.4 Setwise Dependence........................................... 50
3.4.1 Setwise upper orthant and setwise upper set positive de-
pendences ................................................... 50
3.4.2 Setwise association .................................. 51
3.4.3 Setwise dependence by mixture .................. 52
3.4.4 Extension to the setwise negative dependence ......... 53
3.5 Other Approaches............................................. 53
3.6 Positive Dependence Orderings................................ 54
3.6.1 Ordering based on PQD ................................ 54
3.6.2 Conditions on ordering ............................... 55
3.6.3 Ordering defined by PRD............................... 56
3.6.4 Association ordering ................................. 56
3.6.5 PDD-ordering ......................................... 57
3.6.6 Orderings defined from DTP(0,1) and LRD .............. 58
3.6.7 Integral stochastic orderings ........................ 60
3.6.7.1 Supermodular ordering........................ 60
3.6.7.2 Directionally convex ordering................ 61
3.6.8 Generating a family of partial orderings.............. 61
3.7 Bayesian Approach to Stochastic Dependence .................. 63
Chapter 4 Copulas 65
4.1 Introduction ................................................ 65
4.2 Definition and Some Properties .............................. 66
4.3 The Fréchet Bounds........................................... 68
Contents xi
4.3.1 Lower and upper Frechet bounds in the family T{FX, F2)
of bivariate distributions with common marginals ... 68
4.3.2 The Frechet bounds for a copula ....................... 69
4.4 Examples...................................................... 70
4.5 Construction of a Copula...................................... 73
4.5.1 The Riischendorf method................................ 73
4.5.2 Application to polynomial copulas...................... 74
4.5.2.1 Approximation of a copula by a polynomial
copula........................................ 75
4.5.3 Other examples ....................................... 76
4.5.4 Models defined from a distortion function.............. 78
4.5.5 Frailty models......................................... 78
4.5.6 Marshall and Olkin’s generalization.................... 80
4.6 Archimedean Copulas........................................... 84
4.6.1 Definition and basic properties........................ 84
4.6.2 Examples............................................... 85
4.6.3 A characterization of Archimedean copulas.............. 87
4.6.4 The limit of a sequence of Archimedean copulas .... 88
4.6.5 Characterization of Archimedean copulas by their
diagonal copulas....................................... 89
4.6.5.1 Fitting an observed distribution with
an Archimedean copula ........................ 91
4.6.6 Characterization of an Archimedean copula by the cu-
mulative distribution function of Z = C(Uj V) .............. 92
4.6.7 Archimedean copulas with two parameters ............... 94
4.7 Archimax Copulas.............................................. 94
4.7.1 Extreme value distribution and extreme value copula . . 94
4.7.2 Definition of Archimax copulas......................... 95
4.7.3 Construction of bivariate distributions belonging to a
predetermined domain of attraction..................... 95
4.7.4 Examples............................................... 96
4.8 Copulas with Discontinuity Constraints........................ 96
4.8.1 Piecewise additive copulas ............................ 97
4.8.2 Piecewise quadratic copulas ........................... 98
4.8.3 Quadratic copulas with holes........................... 98
4.8.3.1 Admissible rectangles ........................ 99
4.8.3.2 The squeeze algorithm........................ 100
Contents
xii
4.9 Copulas with More than Two Variables......................... 101
4.9.1 m-dimensional Archimedean copulas .................... 102
4.9.1.1 An application............................... 102
4.9.2 Generation of a 3-dimensional copula from its 2-dimensional
marginals ............................................ 105
4.9.2.1 Compatibility of marginals .................. 105
4.9.2.2 Truncation invariance ....................... 105
4.9.3 Linkages.............................................. 108
4.10 Simulation Procedures ....................................... 110
4.10.1 The general case...................................... 110
4.10.2 Archimedean copulas .................................. 110
4.10.3 Archimax distributions ............................... Ill
4.10.4 Marshall and Olkin’s mixture of distributions .........112
4.10.5 Three-dimensional copulas with truncation invariance . 112
Chapter 5 Farlie-Gumbel-Morgenstern Models
of Dependence 113
5.1 Introduction................................................. 113
5.2 Initial Definition........................................... 114
5.3 Regression and Correlation................................... 115
5.4 Iterations................................................... 117
5.5 Dependence Properties........................................ 119
5.6 A Class of n-variate FGM Distributions....................... 119
5.6.1 A class of bivariate FGM distributions with Weibull
marginal distributions................................ 122
5.6.2 A class of three-variate distributions with Weibull
marginal distributions................................ 125
5.6.3 FGM n-variate distributions with Weibull marginals . . 129
5.7 Further Extensions........................................... 130
5.7.1 Huang and Kotz extensions............................. 131
5.7.2 Sarmanov’s extension.................................. 132
5.7.3 Sarmanov-Lee extension................................ 134
5.7.4 Bairamov-Kotz extensions............................. 135
5.7.5 Lai and Xie extension................................. 136
5.7.6 Bairamov-Kotz-Bekgi generalization.................... 137
5.7.7 Concomitants of order statistics ..................... 139
5.8 FGM Sequences................................................ 141
Contents
xiii
Chapter 6 Global Versus Local Dependence between
Random Variables 149
6.1 Introduction ................................................... 149
6.2 Global Measures of Dependence ........................... 150
6.2.1 Desirable properties of a measure of dependence .... 150
6.2.2 Covariance, Q-covariance................................ 151
6.2.3 The coefficient of linear correlation p .............. 153
6.2.3.1 The case when (X, Y) is bivariate normal. . . 153
6.2.3.2 Correlation and extremal properties of normal
distributions ................................. 155
6.2.3.3 p and the moment of inertia around the line
Di : {y = x} ................................ 155
6.2.3.4 A geometric interpretation .................... 156
6.2.3.5 p and concepts of dependence.................. 157
6.2.4 The ps of Spearman and its connection with the
PQD concept ............................................ 157
6.2.4.1 A geometric interpretation of ps ............. 158
6.2.4.2 Estimation of ps .............................. 158
6.2.5 Schweizer-WolfTs index of dependence ................... 159
6.2.6 The Kendall r and its connection with LRD property . 159
6.2.6.1 Estimation of r................................ 160
6.2.7 The Blomqvist medial coefficient........................ 161
6.2.8 r, ps, /? and ordering on the distributions ............ 161
6.2.9 Constructing other global measures ..................... 162
6.2.10 Indices for more than two variables .................... 162
6.2.11 Mutual information, relative entropy and derivatives
measures................................................ 163
6.2.11.1 Definitions.................................... 163
6.2.11.2 Examples ...................................... 165
6.2.11.3 Lin’s measure of association................... 167
6.2.11.4 Zografos’s measure of association.............. 168
6.3 Local Indices of Dependence .................................. 171
6.3.1 Motivation.............................................. 171
6.3.2 Local definition of the dependence...................... 171
6.3.3 Local ps and r ......................................... 172
6.3.4 Local correlation coefficient of Bjerve and Doksum . . 172
6.3.4.1 Motivation and historical remarks ............. 172
6.3.4.2 Definition, properties and limits ............. 173
XIV
Contents
6.3.4.3 Estimations and properties of the estimators 174
6.3.5 Correlation ratio.................................... 175
6.3.6 Local dependence function of Bairamov and Kotz . . . 175
6.3.7 Measures of the tail dependence ..................... 177
6.3.8 Several local indices applicable in survival analysis ... 178
6.3.8.1 The covariance function of Prentice and Cai . 178
6.3.8.2 The conditional covariance rate of Dabrowska 179
6.3.8.3 0 : the ratio of two conditional hazards . . . 183
6.3.8.4^ Other measures derived from 0.............. 188
6.3.8.5 A local measure of LRD dependence : 7/ . . 189
6.4 Non-parametric Estimation of Local Indices.................. 190
6.4.1 The univariate case : estimation of H(x) ............ 190
6.4.2 Bivariate case and conditional hazards .............. . 191
6.4.2.1 Consistency and asymptotic normality of the
estimate of h(x, y) .......................... 192
6.4.2.2 Consistency and asymptotic normality of the
two conditional hazards ...................... 193
6.4.3 Estimation of the indices l and 0 194
6.5 A Search for the Localisation of the Maximal Association . . . 194
6.5.1 Lower and upper tail dependence for the three distribu-
tions ....................................................... 196
6.5.2 0 and the remaining dependence ......................... 197
6.5.3 Index 7 and the instantaneous dependence................ 199
6.5.4 Simulation of the survival bivariate distributions and es-
timation of 0 200
Bibliography 203
Index 217
|
| any_adam_object | 1 |
| author | Drouet Mari, Dominique Kotz, Samuel 1930-2010 |
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| id | DE-604.BV013962647 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T10:56:40Z |
| institution | BVB |
| isbn | 1860942644 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009555886 |
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| physical | XIV, 219 S. graph. Darst. |
| publishDate | 2001 |
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| spellingShingle | Drouet Mari, Dominique Kotz, Samuel 1930-2010 Correlation and dependence Correlatieanalyse gtt Corrélation (Statistique) Dépendance (Statistique) Inferência estatística larpcal Correlation (Statistics) Dependence (Statistics) Korrelation (DE-588)4165343-9 gnd Stochastische Abhängigkeit (DE-588)4220425-2 gnd |
| subject_GND | (DE-588)4165343-9 (DE-588)4220425-2 |
| title | Correlation and dependence |
| title_auth | Correlation and dependence |
| title_exact_search | Correlation and dependence |
| title_full | Correlation and dependence Dominique Drouet Mari ; Samuel Kotz |
| title_fullStr | Correlation and dependence Dominique Drouet Mari ; Samuel Kotz |
| title_full_unstemmed | Correlation and dependence Dominique Drouet Mari ; Samuel Kotz |
| title_short | Correlation and dependence |
| title_sort | correlation and dependence |
| topic | Correlatieanalyse gtt Corrélation (Statistique) Dépendance (Statistique) Inferência estatística larpcal Correlation (Statistics) Dependence (Statistics) Korrelation (DE-588)4165343-9 gnd Stochastische Abhängigkeit (DE-588)4220425-2 gnd |
| topic_facet | Correlatieanalyse Corrélation (Statistique) Dépendance (Statistique) Inferência estatística Correlation (Statistics) Dependence (Statistics) Korrelation Stochastische Abhängigkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009555886&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT drouetmaridominique correlationanddependence AT kotzsamuel correlationanddependence |
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