Correlation and dependence:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London
Imperial College Press
2004
|
| Edition: | Repr. |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013168349&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013168349&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | XIV, 219 S. graph. Darst. |
| ISBN: | 1860942644 |
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| 245 | 1 | 0 | |a Correlation and dependence |c Dominique Drouet Mari ; Samuel Kotz |
| 250 | |a Repr. | ||
| 264 | 1 | |a London |b Imperial College Press |c 2004 | |
| 300 | |a XIV, 219 S. |b graph. Darst. | ||
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Record in the Search Index
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| adam_text | Contents
Preface
vii
Chapter
1
Notations and Definitions
1
1.1
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
Moie
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,l) 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
Préchet
bounds in the family T{Fi
,
F2
)
of bivariate distributions with common marginals
... 68
4.3.2
The
Fréchet
bounds for a copula
............. 69
4.4
Examples
.............................. 70
4.5
Construction of a Copula
..................... 73
4.5.1
The
Rüschendorf
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(U, 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
xii Contents
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
З
-dìmensional
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
η
-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
η
-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-Bekçi
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
ρ
.......... 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
ρ
and the moment of inertia around the line
D1:{y = x)
.................. 155
6.2.3.4
A geometric interpretation
.......... 156
6.2.3.5
ρ
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-
Wolff s 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
τ,
ps,
β
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
Doksům
. . 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
Dąbrowska
179
6.3.8.3
θ
:
the ratio of two conditional hazards
. . . 183
6.3.8.4
Other measures derived from
θ
........ 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 I and
θ
............ 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
θ
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
Θ
....................... 200
Bibliography
203
Index
217
The concept of dependence permeates the Earth and its inhabitants
n a rnost profound manner. Examples of interdependent
meteorological phenomena in nature and interdependence
n
^е
mec ca
social and political aspects of our existence,
not to mention the economic structures, are too
numerous to be cited individually. Moreover, the
dependence is obviously not deterministic but of a stochastic nature. However,
it seems that none of the departments of statistics, engineering, economics
and mathematics in the academic institutions throughout the world offer courses
dealing with dependence concepts and measures.
This book can thus be viewed as an attempt to remedy the situation, and it has been written for
a graduate course or a seminar on correlation and dependence concepts and measures. A modest
background in mathematical statistics and probability and integral calculus is required. The book
is not a full-scale expedition up another statistical Alp. Rather, it is a tour over a somewhat
neglected but important terrain. The chapter on correlation is written for a layman.
|
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| id | DE-604.BV019843499 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T12:07:05Z |
| institution | BVB |
| isbn | 1860942644 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013168349 |
| oclc_num | 177313882 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-824 DE-355 DE-BY-UBR DE-384 DE-83 DE-19 DE-BY-UBM |
| owner_facet | DE-91G DE-BY-TUM DE-824 DE-355 DE-BY-UBR DE-384 DE-83 DE-19 DE-BY-UBM |
| physical | XIV, 219 S. graph. Darst. |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Imperial College Press |
| record_format | marc |
| spellingShingle | Drouet Mari, Dominique Kotz, Samuel 1930-2010 Correlation and dependence 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 | Korrelation (DE-588)4165343-9 gnd Stochastische Abhängigkeit (DE-588)4220425-2 gnd |
| topic_facet | Korrelation Stochastische Abhängigkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013168349&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013168349&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT drouetmaridominique correlationanddependence AT kotzsamuel correlationanddependence |
Table of Contents
Order paper/chapter scan
Order paper/chapter scan
Branch Library Mathematics & Informatics
| Call Number: |
0102 MAT 628f 2005 A 4122(1,2004)
Floor plan |
|---|---|
| Copy 1 | Available for loan On Shelf |