| Titre : | Estimating the mean of heavy tailed distribution under random truncation |
| Auteurs : | Khanssa BEN DAHMANE, Auteur ; Fatah Benatia, Directeur de thèse |
| Type de document : | Monographie imprimée |
| Editeur : | Biskra [Algérie] : Faculté des Sciences Exactes et des Sciences de la Nature et de la Vie, Université Mohamed Khider, 2022 |
| Format : | 1 vol. (103 p.) / couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Mots-clés: | Asymptotically normality, Extreme value Theory, Extreme value index, Lynden-Bell estimator, Random variation, Heavy-tails, Random truncation |
| Résumé : |
The main aim of this thesis is to deploy and develop a new estimator for the mean that is based on the famous paper by Peng, 2001. Our case focuses on dealing with data when it becomes incomplete with a particular interest in the case of right-truncated, an asymptotic estimator is proposed and its behavior examined in a simulation study. We treat throughout our study two branches: Survival Analysis and Extreme Value Theory which has emerged as one of the most important statistical disciplines for the applied sciences over the last 50 years. The rst objective of this thesis is to collect and simplify what has been done in the study of extreme values theory. This branch is interested in rare events and the causes of all disasters we know and of all economic crises. In addition, the second objective is to present an introduction that is devoted to the basic notions of survival analysis. Furthermore, we present two cases of incomplete data (censored and truncated) with giving the non-parametric estimator of the mean for each case.The main aim of this thesis is to deploy and develop a new estimator for the mean that is based on the famous paper by Peng, 2001. Our case focuses on dealing with data when it becomes incomplete with a particular interest in the case of right-truncated, an asymptotic estimator is proposed and its behavior examined in a simulation study. We treat throughout our study two branches: Survival Analysis and Extreme Value Theory which has emerged as one of the most important statistical disciplines for the applied sciences over the last 50 years. The rst objective of this thesis is to collect and simplify what has been done in the study of extreme values theory. This branch is interested in rare events and the causes of all disasters we know and of all economic crises. In addition, the second objective is to present an introduction that is devoted to the basic notions of survival analysis. Furthermore, we present two cases of incomplete data (censored and truncated) with giving the non-parametric estimator of the mean for each case. |
| Sommaire : |
Abstract iii Dedication v Acknowledgments vii Appendix A: Abbreviations and Notations ix List of Figures xiii List of Tables xv Contents xvii 1 Introduction 1 I Preliminary Theory 5 2 Extreme Value theory 7 2.1 Order statistics . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Empirical distribution function and order statistics 9 2.1.2 Distribution function and density of the maximum 9 2.1.3 Upper end point . . . . . . . . . . . . . . . . . . . 10 2.1.4 Quantile function . . . . . . . . . . . . . . . . . . 10 2.1.5 Empirical quantile function . . . . . . . . . . . . 11 2.1.6 Tail quantile and emperical tail quantile function 11 2.1.7 Distributions of order statistics . . . . . . . . . . 12 2.2 Distribution of extreme values . . . . . . . . . . . . . . . 13 2.3 Limit distributions . . . . . . . . . . . . . . . . . . . . . . 14 xvii2.4 Generalized extreme values distributions (GEVD) . . . . 16 2.5 Generalized Pareto Distribution (GPD) . . . . . . . . . . 19 2.6 Regularly Varying distributions . . . . . . . . . . . . . . 22 2.6.1 First Order Regular variation Assumption . . . . 24 2.6.2 Second Order Regular variation Assumption . . . 25 2.6.3 Third Order Regular variation Assumption . . . . 26 2.7 Domain of attraction . . . . . . . . . . . . . . . . . . . . 26 2.8 Estimation of the extreme value index . . . . . . . . . . . 32 2.8.1 Pickand’s estimator . . . . . . . . . . . . . . . . . 32 2.8.2 Hill’s estimator . . . . . . . . . . . . . . . . . . . 34 2.8.3 Optimal sample fraction selection . . . . . . . . . 35 3 Survival Analysis 39 3.1 Basic concepts and denitions . . . . . . . . . . . . . . . 39 3.2 Laws of large numbers . . . . . . . . . . . . . . . . . . . 43 3.3 Estimating the mean of a heavy-tailed distribution . . . . 46 3.3.1 Estimating the mean of a heavy-tailed distribution in the case of nite second moment . . . . . . . . 46 3.3.2 Estimating the mean of a heavy-tailed distribution in the case of innite second moment . . . . . . . 47 3.3.3 Kernel-type estimator of the mean for a heavy tailed distribution . . . . . . . . . . . . . . . . . . 50 4 Taxonomy of incomplete Data 55 4.1 Censoring . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.1.1 Types of Censoring Mechanisms . . . . . . . . . 56 4.1.2 Estimation under random right-censoring . . . . 61 4.2 Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2.1 Right truncation . . . . . . . . . . . . . . . . . . 65 4.2.2 Left truncation . . . . . . . . . . . . . . . . . . . 65 4.2.3 Interval truncation . . . . . . . . . . . . . . . . . 66 4.3 Estimation under random truncated model . . . . . . . . 66 4.3.1 Estimation the distribution function under truncation model . . . . . . . . . . . . 66 xviii4.3.2 Estimation Tail-index under truncation model . . 69 II Main results 71 5 Estimating the mean of a heavy tailed distribution under random truncation 73 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 The assumptions and the main results . . . . . . . . . . . 78 5.3 Simulation study . . . . . . . . . . . . . . . . . . . . . . . 79 5.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.4.1 Proof of Theorem 2.1 . . . . . . . . . . . . . . . 83 5.4.2 Proof of Corollary 2.1 . . . . . . . . . . . . . . . 90 Conclusions & Outlook 93 Bibliography 95 Appendix B: Soware R 101 List of Publications and Communication 103 5.1 Articles in Refereed Journals . . . . . . . . . . . . . . . . 103 5.2 Communications . . . . . . . . . . . . . . . . . . . . . . . 103 |
| En ligne : | http://thesis.univ-biskra.dz/5686/1/THESIS.DOC.BENDAHMANE.pdf |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/120 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
