Thesis Contributions to simulation-based estimation methods ORSO, Samuel Abstract The focus of this thesis is twofold. First, it delivers a new look at existing simulation-based methods for statistical inference in parametric problems. Emphasis is placed on finite sample theoretical properties and computational efficiency. In particular, a simple and computationally efficient method for inference is proposed. It is shown that exact inference may be claimed in theory in some situations even though sample size is kept fixed. Numerical examples demonstrate the wide applicability of this method. Second, a general class of flexible models for dependent random phenomena is studied. Emphasis is placed on problems of point estimations due to the presence of outliers or because of the underlying computational burden. To tackle these issues, a new multi-step robust and computationally efficient estimator is proposed. Asymptotic properties are studied along with illustrative examples. Reference ORSO, Samuel. Contributions to simulation-based estimation methods. Thèse de doctorat : Univ. Genève, 2019, no. GSEM 66 DOI : 10.13097/archive-ouverte/unige:121536 URN : urn:nbn:ch:unige-1215361 Available at: http://archive-ouverte.unige.ch/unige:121536 Disclaimer: layout of this document may differ from the published version. 1 / 1 Contributions to simulation-based estimation methods by Samuel Orso A thesis submitted to the Geneva School of Economics and Management, University of Geneva, Switzerland, in fulfillment of the requirements for the degree of PhD in Statistics Members of the thesis committee: Prof. Maria-Pia Victoria-Feser, Co-advisor, University of Geneva Prof. St´ephane Guerrier, Co-advisor, Pennsylvania State University Prof. Stefan Sperlich, Chair, University of Geneva Prof. Yanyuan Ma, Pennsylvania State University Thesis No. 66 January 2019 La Facult´ed’´economie et de management, sur pr´eavis du jury, a autoris´el’impression de la pr´esente th`ese, sans entendre, par-l`a, ´emettre aucune opinion sur les propositions qui s’y trouvent ´enonc´ees et qui n’engagent que la responsabilit´ede leur auteur. Gen`eve, le 29 janvier 2019 Le doyen, Marcelo OLARREAGA Impression d’apr`es le manuscrit de l’auteur. Acknowledgements This thesis has greatly benefitted from the academic and moral support of many, and as hard as it seems, I would like to express my gratitude to all of them. Foremost, I would like to thank my two supervisors Prof. Maria-Pia Victoria-Feser and Prof. St´ephane Guerrier. I would not be writting these lines if it was not for them. They made me discover statistics as an under-graduate student in business administration and have never let me down ever since. I am not sure I would have had their patience with someone who had such a poor background. St´ephane has been very influencial in my decisions to undertake studies in statistics and to pursue a PhD. He has always supported me and is very inspirational. Maria-Pia has never stopped me for trying my own way for research, better, she made it possible regardless of the circumstances. I admire her commitment and all the positive energies she puts into work. I would like to thank Prof. Stefan Sperlich, the chair of my jury, for the interest he took in my work, for his constructive feedbacks and for his professionalism. Likewise, I would like to thank Prof. Yanyuan Ma who honoured me by accepting to be my external expert and whose instructive comments improved the content of this thesis. I would also like to thank my incredible colleagues. I am forever indebted to Marco, a.k.a. Marc-small-oh, and Marc-Olivier, a.k.a. Marc-Big-Oh, for their valuable discus- sions from which this thesis has benefitted. I shared unforgettable moments with the “dˆıner de cons” team: Kustrim, Elise, Rose, Marc-Olivier, Mattia and Dany. I enjoyed sportive time with the “badminton” team: Rami, Walid, Mark, Haotian, Mattia and Dany. Throughout these years, I have always loved sharing ideas with my colleagues at the GSEM. Among them, I would like to acknowledge: Jean-Christophe, Jonathan, Ozgu, Roberto, Sebastian, Justine, Ingrid, Steffen, Mark, Pierre-Yves, Linda, Setareh, Laura, Guillaume, Cesare, Julien, Alban, Jiajun, Benjamin, Kasia, Mucyo, Yuming et al. My office mates Haotian and Ga¨etan were very supportive all along my dissertation and I specially want to thank them. I would like to thank my parents, Charly and Corine, my three sisters and their partners, Jo¨elle, Lucien, Myriam, Estelle and Sergio, for their love and for always being supportive. A special though goes to my two lovely little nieces Anouk and Justine. This thesis would have been impossible without the support of my in-laws Jordan, Margarita and Aleks. On a great deal of occasions, Margarita has volunteered to compensate my lack of availability to my family. This thesis was one of the last thing, if not the last, that I started before my son Isaac was born. He encouraged me a lot for example by saying: “Mon papa fait des statistiques” 1, or with the many drawings that covers half of my wall at the office. My daughter Tania was born later. Her arrival gave me the courage to attack the last marathon that represents the writing of a thesis. Her good sleeps were the encouragement I needed. Many of the ideas that came up to me during the thesis 1French for: “My dad does statistics” ii happened during the bedtime. These little reflections of myself have greatly impacted this thesis. Finally, I would like to thank my soulmate and wife Linda for all these years of sharing and giving. She shared my ups and down. She never stopped believing in me. She gave me all the forces that leads me to say today: “I have Ph.inisheD”. Abstract The focus of this thesis is twofold. First, it delivers a new look at existing simulation- based methods for statistical inference in parametric problems. Emphasis is placed on finite sample theoretical properties and computational efficiency. In particular, a simple and computationally efficient method for inference is proposed. It is shown that exact inference may be claimed in theory in some situations even though sample size is kept fixed. Numerical examples demonstrate the wide applicability of this method. Second, a general class of flexible models for dependent random phenomena is studied. Emphasis is placed on problems of point estimations due to the presence of outliers or because of the underlying computational burden. To tackle these issues, a new multi-step robust and computationally efficient estimator is proposed. Asymptotic properties are studied along with illustrative examples. R´esum´e Cette th`ese comporte deux parties. Premi`erement, elle d´elivre un nouveau regard sur diff´erentes m´ethodes par simulations d´evelopp´ees pour faire de l’inf´erence statistique sur des probl`emes param´etriques. Le focus est port´esur les propri´et´es th´eoriques en ´echantillon fini et les probl`emes computationnelles. En particulier, une m´ethode simple et computationnellement ´efficiente est propos´ee. Il est d´emontr´equ’il est possible d’obtenir une inf´erence exacte dans certaines situations tout en gardant la taille d’´echantillon fixe. Des examples num´eriques illustrent le vaste champ d’application de cette m´ethode. Deuxi`emement, une classe g´en´erale de mod`eles pour des variables al´eatoires d´ependantes est ´etudi´e. Les sujets principaux sont les probl`emes d’estimation ponctuelles dˆus `a la pr´esence de donn´ees aberrantes ou `acause de probl`emes computationnelles. Afin d’adresser ces questions, un nouvel estimateur robuste et computationnellement efficient est propos´e. Ses propri´et´es asymptotiques sont ´etudi´ees. Des examples viennent illustrer ces r´esultats. Contents Acknowledgements i Abstract iii R´esum´e v Introduction 1 List of notation 3 1 SwiZs: Switched Z-estimators 5 1.1 Introduction ................................... 5 1.2 Setup ....................................... 7 1.3 Equivalent methods ............................... 8 1.4 Exact frequentist inference in finite sample .................. 15 1.5 Asymptotic properties ............................. 20 1.6 Examples .................................... 25 1.7 Simulation study ................................ 33 Appendices 67 1.A Technical results ................................ 67 1.B Finite sample .................................. 68 1.C Asymptotics ................................... 72 1.D Additional simulation results .......................... 76 1.D.1 Lomax distribution ........................... 76 1.D.2 Random intercept and random slope linear mixed model ...... 83 1.D.3 M/G/1 queueing model ........................ 88 2 Bounded-Influence Robust Estimation of Copulae 91 2.1 Introduction ................................... 91 2.2 Framework .................................... 93 2.3 Asymptotic results ............................... 95 2.4 Some practical aspects for indirect inference procedure ...........100 2.4.1 Point estimator .............................100 2.4.2 Inference .................................101 2.5 Bounding the Influence Function .......................102 2.6 A weighted maximum likelihood indirect estimator .............104 2.7 Simulation Study ................................104 2.7.1 Bivariate Clayton copula ........................105 viii Contents 2.7.2 Factor Copula ..............................110 2.8 Application to income mobility ........................115 Appendices 131 2.A Intermediate results ...............................131 2.B Main proofs ...................................132 2.C Additional results