| Titre : | MULTI-PARAMETRIC COPULA ESTIMATION BASED ON MOMENTS METHOD UNDER CENSORING |
| Auteurs : | Nesrine IDiou, 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. (121 p.) / couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Mots-clés: | Copula, Archimedean copulas models, Semi-parametric estimation, Moments method, Survival copula, Right censored data, Frailty mode |
| Résumé : |
This thesis combines two interesting branches of statistics: survival analysis and copula theory. The primary objective is to extend the copula theory results via semi-parametric estimation, under censored data. More precisely, we are interested by a copulas semi-parametric estimation, based on the classical moments estimation method, adapted for bivariate censored data. There are various kinds of censoring, we are only look at doubly and singly right-censored data. As theoretical results, general formulas were proved with analytical forms of the obtained estimators. According to early research, many asymptotic results obtained in the framework of non-parametric statistics for right-censored observations are based on the Kaplan Meier estimator, which estimates the survival function. Taking into account the results of Lopez and Saint-Pierre (2012) [72], Gribkova and Lopez (2015) [39], the asymptotic normality of the empirical survival copula was established for the two cases of censoring. The dependence structure between the bivariate survival times was modeled under the assumption that the underlying copula is Archimedean. Accounting for various censoring patterns (singly or doubly censored), a simulation study was performed efficiency and robustness of the new estimator proposed. Individual random parameters, which are commonly understood as frailty parameters, are another tool frequently employed for modeling multivariate survival data. We implemented this model for two-variable survival data using Archimedean copulas in the final part of the thesis. The frailty variables considered here are latent variables that are not observed, are nevertheless one-dimensional. In the example presented, this variable characterized the effect of the individual on the recurrence time. Then we looked at Clayton-Oakes copulas in particular, and even the model with gamma-type frailty. For each of these two models, the copulas used for the bivariate survival functions are the same. Even so, the marginal survival functions are modeled in different ways. The applications for health-related survival data were next examined. |
| Sommaire : |
Scientific contributions iv Contents vi List of Figures ix List of Tables x Introduction 1 1 Preliminary 6 1.1 Foundations definition . . . . . . . . . . . . . . . . . . . . 7 1.2 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Order statistics . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.2 H-volume notion, 2-increasing functions . . . . . . . . . 9 2 Copula conseptions 11 2.1 Bivariate Copula . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Copula and Sub-Copula . . . . . . . . . . . . . . . . . . 13 2.1.2 Sklar’s Theorem . . . . . . . . . . . . . . . . . . . . . . 15 2.1.3 Copulas and random variables . . . . . . . . . . . . . . . 17 2.1.4 Fréchet-Hoeffding Boundaries . . . . . . . . . . . . . . . 19 2.1.5 Survival and semi-survival copulas . . . . . . . . . . . . 21 2.1.6 Copula properties . . . . . . . . . . . . . . . . . . . . . 22 2.2 Bivariate Copula families . . . . . . . . . . . . . . . . . . . 24 2.2.1 Usual Copulas . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.2 Archimedean Copulas . . . . . . . . . . . . . . . . . . . 26 2.2.3 Extreme values Copulas . . . . . . . . . . . . . . . . . . 28 2.2.4 Bivariate extreme values distributions . . . . . . . . . . . 30 2.3 Multivariate Copula . . . . . . . . . . . . . . . . . . . . . . 31 2.3.1 Sklar’s theorem . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.2 A multivariate copula’s properties . . . . . . . . . . . . . 33 2.3.3 Multivariate parametric copula . . . . . . . . . . . . . . 33 3 Copula and dependence 37 3.1 Association measures . . . . . . . . . . . . . . . . . . . . . 38 3.1.1 Concordance measures . . . . . . . . . . . . . . . . . . . 38 3.1.2 Kendall’s Tau . . . . . . . . . . . . . . . . . . . . . . . . 41 vi3.1.3 Spearman’s Rho . . . . . . . . . . . . . . . . . . . . . . 42 3.2 Dependence measure . . . . . . . . . . . . . . . . . . . . . . 43 3.2.1 Tail dependency . . . . . . . . . . . . . . . . . . . . . . 44 4 Survival Analysis and Copulas 46 4.1 Survival time notion . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Incomplete data . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2.1 Truncated notion . . . . . . . . . . . . . . . . . . . . . . 48 4.2.2 Censoring notion . . . . . . . . . . . . . . . . . . . . . . 49 4.3 Semi-parametric estimation for Copula models . . . . . 51 4.3.1 Maximum Likelihood Estimation (MLE) . . . . . . . . . 51 4.3.2 Margin Inference Function Method (IFM) . . . . . . . . . 52 4.3.3 The Pseudo-maximum likelihood method (PML) . . . . . 53 4.3.4 Moments Estimation method based on Kendall’s Tau and Spearman’s Rho . . . . . . . . . . . . . . . . . . . . . . 53 4.4 Non-parametric estimation for right-censoring model 55 4.4.1 Kaplan-Meier Estimator . . . . . . . . . . . . . . . . . . 55 4.4.2 Kernel density estimator . . . . . . . . . . . . . . . . . . 56 4.5 Non-parametric estimation for mixed censoring model 56 4.5.1 The Patilea and Rolin Estimator . . . . . . . . . . . . . . 57 5 A semi-parametric estimation of copula models under right-censoring 58 I A semi-parametric estimation of copula models based on moments methods under right-censoring 59 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.3 Moments estimator for right-censoring . . . . . . . . . 66 5.4 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . 69 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 II Survival Copula parameters estimation for Archimedean family under singly censoring 76 5.7 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.8 Important results . . . . . . . . . . . . . . . . . . . . . . . . 79 5.9 Parameters estimation under singly right censored variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.10 Application: illustrative examples . . . . . . . . . . . . . 83 5.11 Simulation studies . . . . . . . . . . . . . . . . . . . . . . . 84 5.12 Application to a Real Data Set . . . . . . . . . . . . . . . 89 5.13 Conclusion and perspective . . . . . . . . . . . . . . . . . 90 vii5.14 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6 Copulas and frailty models in multivariate survival data 92 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2 Survival models . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.3 Copula models . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3.1 Example: Clayton model . . . . . . . . . . . . . . . . . . 97 6.4 Frailty model . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.4.1 Bivariate survival copula and frailty model . . . . . . . . 100 6.4.2 Clayton-Oakes copula and gamma frailty model . . . . . 101 6.5 Application to hemodialysis data . . . . . . . . . . . . . . 102 6.6 Conclusion and perspectives . . . . . . . . . . . . . . . . . 104 6.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Bibliography 107 |
| En ligne : | http://thesis.univ-biskra.dz/5685/1/These%20IDIOU%20Nesrine%202022.pdf |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/119 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
