| Titre : | ON THE CONSISTENCY AND THE ASYMPTOTIC NORMALITY OF SEVERAL CONDITIONAL MODELS, DEPENDENT CASE |
| Auteurs : | Seba Djillali, Auteur ; Rahman Saâdia, Directeur de thèse |
| Type de document : | texte manuscrit |
| Editeur : | Université de Saida - Dr Moulay Tahar. Faculté des Sciences. Département de Mathématiques., 2018/2019 |
| Format : | 67ص |
| Accompagnement : | CD |
| Langues: | Anglais |
| Index. décimale : | BUC-M 008350 |
| Catégories : |
Master Mathématiques:Analyse stochastique, statistique des processus et applications (ASSPA) |
| Résumé : |
There are many situations in which we study the link between two variables in
order to be able to predict new values of one of them given the other one, this problem occurs with real, multivariate variables and functional variables. There are several ways to approach the prediction setting, in this dissertation we have been interested in two important models: conditional mode and conditional quantile which are studied when the explanatory variables are functional and the response variable still real. We have provided some theoretical supports by showing how the dependence is acting on the asymptotic behavior of the non-parametric functional method. In the last chapter, we have illustrated an application in which we have applied the conditional quantile approach in time-series analysis to the prediction and the building of confidence bands, then we have implement our methodology with el Nino data which is a real data study that test the performance of the conditional quantile estimator. |
| Note de contenu : |
Contents
1 General introduction 5 1.1 Bibliographic context . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.2 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.3 Brief outline of the dissertation . . . . . . . . . . . . . . . . . 13 2 Functional Conditional Mode 14 2.1 The model and its estimator . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Assumptions and some remarks . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Assumptions and notations . . . . . . . . . . . . . . . . . . . 16 2.2.2 Remarks on the assumptions . . . . . . . . . . . . . . . . . . . 17 2.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.1 Consistency of the estimator . . . . . . . . . . . . . . . . . . . 19 2.3.2 Asymptotic normality . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Technical proofs and auxiliary results . . . . . . . . . . . . . . . . . 23 2.4.1 Proof of Proposition 2.1 . . . . . . . . . . . . . . . . . . . . . 23 2.4.2 Proof of Proposition 2.2 . . . . . . . . . . . . . . . . . . . . . 25 2.4.3 Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . 28 3 Functional Conditional Quantile 36 3.1 The model and its estimator . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Assumptions and some remarks . . . . . . . . . . . . . . . . . . . . . 38 3.2.1 Assumptions and notations . . . . . . . . . . . . . . . . . . . 38 3.2.2 Remarks on the assumptions . . . . . . . . . . . . . . . . . . . 38 3 CONTENTS 3.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.1 Consistency of the estimator . . . . . . . . . . . . . . . . . . . 39 3.3.2 Asymptotic normality . . . . . . . . . . . . . . . . . . . . . . 40 3.4 Technical proofs and auxiliary result . . . . . . . . . . . . . . . . . . 42 3.4.1 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.2 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . 44 3.4.3 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . 45 3.4.4 Proof of Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . 52 4 Application and conclusion 53 4.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.1.1 Confidence bands . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.1.2 Application to prediction . . . . . . . . . . . . . . . . . . . . . 55 4.1.3 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Bibliography 63 4 |
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