Titre : | New approach for estimating the distribution tails for incomplete data |
Auteurs : | SAIDA Mancer, Auteur ; Abdelhakim Necir , Directeur de thèse |
Type de document : | Thése doctorat |
Editeur : | Biskra [Algérie] : Faculté des Sciences Exactes et des Sciences de la Nature et de la Vie, Université Mohamed Khider, 2023 |
Format : | 1 vol. (108 p.) / couv. ill. en coul / 30 cm |
Langues: | Français |
Résumé : |
Our work is situated in the field of extreme values’ statistics for incomplete data namely the truncation and the censoring. In this context, several approaches for estimating distribution tails under random truncation have recently been developed: Gardes & Stupfler (2015) [18], Benchaira et al. (2015) [5], Benchaira et al. (2016a) [6], Benchaira et al. (2016b) [7], et Haouas et al. (2018) [21]. The first objective of this thesis is to define a new method ” the semiparametric method” to estimate the tail index of the distribution, while the majority of the existing method depend on the non-parametric estimator of the tail distribution index such as LyndeBell and Woodroofe, the ours is based on the semi-parametric estimator defined in Wang 1989 [48] that allows us introducing new estimators with high efficiency. For the second objective, at this point, we are interested in correcting the error of kernel estimators, such as Benchaira et al. (2016b)’s estimator, so we have introduced a new kernel estimator with reduced bias at the same time. Without forgetting the complete data, in the third objective of this thesis we add a new estimator of the extreme value’s index beside the well-known estimators such as Hill, Peng, ... etc. The new one is characterized by its robustness and stability and was developed by using the idea which was presented in Basu 1998 [2] based on the density power divergence function |
Sommaire : |
TABLE OF CONTENTS Page List of Tables viii List of Figures x Introduction 1 1 Preliminaries 5 1.1 Extreme value theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.1 Limit law distribution of Maxima . . . . . . . . . . . . . . . . 6 1.1.2 Generalized extreme value distribution . . . . . . . . . . . . 8 1.1.3 Regular variation and Domains of attraction . . . . . . . . . 9 1.1.4 Estimation of the extreme value index . . . . . . . . . . . . . 13 1.2 Incomplete data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.1 Cencoring data . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.2 Truncated data . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.3 Tail index estimation under right truncating data . . . . . . . . . . . 18 1.3.1 Gardes and Stupfler estimator . . . . . . . . . . . . . . . . . . 20 1.3.2 Benchaira et al estimator . . . . . . . . . . . . . . . . . . . . . 20 1.3.3 Worms and Worms estimator . . . . . . . . . . . . . . . . . . . 21 1.3.4 Kernel estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.5 Haouas et al. estimator . . . . . . . . . . . . . . . . . . . . . . 23 2 Semiparametric tail-index estimation for randomly right-truncated heavy-tailed data 24 2.1 Semi-parametric estimator of the truncation distribution function F 25 viTABLE OF CONTENTS 2.2 Construction of the new estimator . . . . . . . . . . . . . . . . . . . . 25 2.3 Main results and Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1 Important Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.2 Theorems and Proofs . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5 Real data example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3 Bias reduction in kernel tail index estimation for randomly truncated Pareto-type data 54 3.1 Bias reduction of γb1,K . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Main results and proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2.1 Instrumental result . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2.2 Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.3.1 Graphical diagnostics . . . . . . . . . . . . . . . . . . . . . . . 64 3.3.2 A heuristic procedure to estimate the tail Index γ1 . . . . . . 66 3.4 Real data example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4 A weighted minimum density power divergence estimator for the Pareto-tail index 87 4.1 Minimum density power divergence . . . . . . . . . . . . . . . . . . . 88 4.2 Weighted MDPD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.3 WMDPD estimation of the tail index . . . . . . . . . . . . . . . . . . . 90 4.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.5 Influence function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.6 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.7 Important lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Conclusion 120 Bibliography 121 |
En ligne : | http://thesis.univ-biskra.dz/id/eprint/6063 |
Disponibilité (1)
Cote | Support | Localisation | Statut |
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TM/139 | Théses de doctorat | bibliothèque sciences exactes | Consultable |