Estimation techniques and goodness-of-fit tests for certain copula classes in large dimensions [Elektronische Ressource] / vorgelegt von Christian Hering

Sujets

• Mathematics

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• Publié par: universitat_ulm
• Publié le: 01 janvier 2011
• Nombre de lectures: 22
• Langue: English
• Poids de l'ouvrage: 4 Mo

Extrait

Estimation Techniques and Goodness-of-fit
Tests for Certain Copula Classes in Large
Dimensions
Christian HeringEstimation Techniques and
Goodness-of-fit Tests for Certain Copula
Classes in Large Dimensions
Dissertation zur Erlangung des Grades eines
Doktors der Naturwissenschaften, Dr. rer. nat.,
der Fakultät für Mathematik und Wirtschaftswissenschaften
an der Universität Ulm
vorgelegt von
Christian Hering
aus Stuttgart
Ulm, im Feburar 2011Amtierender Dekan: Prof. Dr. Paul Wentges
Erstgutachter: Prof. Dr. Ulrich Stadtmüller
Zweitgutachter: Prof. Dr. Rüdiger Kiesel
Tag der Promotion: 15. April 2011
Ich versichere hiermit, dass ich die Arbeit selbständig angefertigt habe und keine
anderen als die angegebenen Quellen und Hilfsmittel benutzt sowie die wörtlich
oder inhaltlich übernommenen Stellen als solche kenntlich gemacht habe.
Ulm, den 14. Feburar 2011 Christian HeringFür meine ElternDas, wobei unsere Berechnungen versagen, nennen wir Zufall.
Albert EinsteinAcknowledgements
First and foremost, I am deeply grateful to Professor Ulrich Stadtmüller for pro-
viding me with the opportunity to prepare this doctoral thesis. His guidance and
innumerable mathematical discussions contributed very much to the results given
in this work. I enjoyed teaching in and being a member of the Institute of Number
Theory and Probability Theory and want to thank Professor Ulrich Stadtmüller and
the director of the Institute of Number Theory and Probability Theory, Professor
Helmut Maier, for giving me this possibility. I am additionally grateful for the finan-
cial support, provided by the Institute of Number Theory and Probability Theory,
enabling me to visit conferences on copula theory and its applications.
I extend my gratitude to Professor Rüdiger Kiesel for being my second advisor and
co-examiner of this doctoral thesis.
I am also indebted to Dr. Marius Hofert, who was always willing to answer my ques-
tions, especially when I started working on copulas. Special thanks go to my friend
Dr. Jan-Frederik Mai for proof-reading this thesis and for discussing and sharing his
thoughts and ideas with me numerous times.
Furthermore, I want to cordially thank my colleagues and friends Matthias Lutz
and Andreas Rupp for fruitful discussions and various coffee breaks which helped to
create a great working atmosphere. I would like to thank Magda Mroz and Michael
Weyhmüller for critically reading and commenting on preliminary drafts of this dis-
sertation. I also thank Stefan Ehrenfried, Dennis Schätz, Patrick Scherer, Leonie
Selinka and Marc Wittlinger for bending my thoughts from mathematics whenever
it was necessary. Moreover, I thank Dr. Hartmut Lanzinger and Dr. Monika Thal-
maier for supporting my teaching efforts.
Last and most importantly, I would like to thank my family. My parents, Hermann
and Edeltraud, and my sister Simone supported me from the beginning and they al-
ways believed in me and in my academic success. I deeply appreciate their enduring
love. Finally, there is the most important person in my life, my wife Eva. Thank
you for your love and your support, which made all this possible.
Ulm, February 2011 Christian Hering
1Contents
Acknowledgements 1
Contents 3
Zusammenfassung 5
1 Abstract 11
2 Mathematical Background 15
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Sampling Strategies . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.3 Estimation Procedures . . . . . . . . . . . . . . . . . . . . . . 22
2.2.4 Goodness-of-fit Techniques . . . . . . . . . . . . . . . . . . . 26
2.2.5 Some Important Copula Classes . . . . . . . . . . . . . . . . 29
2.3 Lévy Subordinators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Empirical Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Exchangeable Archimedean Copulas 43
3.1 Probabilistic Construction and Sampling Strategies . . . . . . . . . . 43
3.1.1 The Probabilistic Model of Marshall and Olkin . . . . . . . . 43
3.1.2 Sampling Based on a Particular Transformation . . . . . . . . 45
3.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1 Estimation Techniques Based on Transformations . . . . . . . 51
3.2.2 Minimum Distance Estimation . . . . . . . . . . . . . . . . . 52
3.2.3 A Strong Consistency Result . . . . . . . . . . . . . . . . . . 54
3.2.4 An Empirical Study and an Application to Real World Data 63
3.3 Goodness-of-fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . 70
3.3.2 A Large Scale Simulation Study . . . . . . . . . . . . . . . . 73
3.3.3 A Graphical Goodness-of-fit Test and a Real World Example 79
4 Nested Archimedean Copulas 85
4.1 Mixtures of Distributions and McNeil’s Sampling Algorithm . . . . . 85
4.2 A New View on Nested Archimedean Copulas . . . . . . . . . . . . . 87
4.2.1 Compatible Generators . . . . . . . . . . . . . . . . . . . . . 88
4.2.2 Probabilistic Construction and Sampling . . . . . . . . . . . . 90
3Contents
4.2.3 Dependence Properties . . . . . . . . . . . . . . . . . . . . . . 93
4.2.4 Examples of New Families . . . . . . . . . . . . . . . . . . . . 98
4.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.3.1 Maximum Likelihood Estimation and Estimation Based on
Margins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.3.2 Iterated Maximum Likelihood Estimation . . . . . . . . . . . 103
5 Marshall-Olkin Copulas 113
5.1 Probabilistic Construction and Sampling Strategies . . . . . . . . . . 113
5.2 Bivariate Maximum Likelihood Estimation . . . . . . . . . . . . . . . 117
5.3 A Method of Moments Approach for Parameter Estimation . . . . . 119
5.3.1 A Motivating Example and Preliminaries . . . . . . . . . . . 121
5.3.2 The Estimation Procedure: Step I . . . . . . . . . . . . . . . 123
5.3.3 The Estimation Procedure: Step II . . . . . . . . . . . . . . . 128
5.3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 132
6 Conclusion and Outlook 135
Appendix 139
A A Word Concerning the Implementation . . . . . . . . . . . . . . . . 139
B Proofs for Identifiability . . . . . . . . . . . . . . . . . . . . . . . . . 139
C Identifiability Condition for Used Archimedean Families . . . . . . . 142
D R-code for the Implemented Sampling Algorithm for nAC . . . . . . 145
E R-code for Estimating nAC . . . . . . . . . . . . . . . . . . . . . . . 146
F R-code for the Implemented Sampling Algorithm for eMO . . . . . . 147
List of Figures 149
List of Tables 151
Bibliography 153
4