| Titre : | Estimation Of Continuous Time Semi Markov Process |
| Auteurs : | Boubred Maroua, Auteur ; Mokhtar Fatiha, Directeur de thèse |
| Type de document : | texte manuscrit |
| Editeur : | Université de Saida - Dr Moulay Tahar. Faculté des Sciences. Département de Mathématiques., 2019/2020 |
| Format : | 93ص |
| Accompagnement : | CD |
| Langues: | Français |
| Index. décimale : | BUC-M 008334 |
| Catégories : |
Analyse stochastique, statistique des processus et applications (ASSPA) |
| Résumé : |
In this work, we explained the continuous-time semi-Markov model with
a discrete set of states. We defined empirical estimators of important quanti- ties such as semi-Markov kernel, sojourn time distributions, transition prob- abilities, and hazard rate function. We gave results about their asymptotic properties. The present work aims at the introduction of the continuous-time semi- Markov model as a candidate model for the description of asthma control, Tunisia, and Algeria Coronavirus data. For asthma control, it was very important to study this data with covariate variable (BMI), using the Wald test, we can conclude the decreasing or increasing effects of this variable. The process of Algeria Coronavirus data was represented with two sta- tistical models Markov and semi Markov model and with the parametric and nonparametric methods. Semi-Markov package in R Language was used for the implementation of the parametric-method however, for the nonpara- metric one, we had developed our functions. Parametric methods provide estimators with several attractive asymptotic properties; however, these es- timators present inconvenience when the sample size is small. Since appli- cations of parametric methods presuppose certain conditions concerning the sample size, this difficulty could be affected through the application of non- parametric methods. For the hazard rate functions, the semi-Markov process maybe a better fit for the previous model. For providing more accurate forecasting results for Algeria Coronavirus data one more ways the accessibility into instantaneous results about Coron- avirus cases and the inclusion of different covariate variables like age, chronic diseases,· · · |
| Note de contenu : |
Contents
Acknowledgments 3 Notations 6 Introduction 9 1 Introduction and preliminaries 12 1.1 Definitions and theorems . . . . . . . . . . . . . . . . . . . . 12 1.2 Discrete-time Markov chain . . . . . . . . . . . . . . . . . . . 15 1.2.1 State classification . . . . . . . . . . . . . . . . . . . . 17 1.3 Continuous-time Markov chain . . . . . . . . . . . . . . . . . . 18 2 Discrete-time semi-Markov process 20 2.1 Markov renewal chains and semi-Markov chains . . . . . . . . 20 2.2 Elements of statistical estimation . . . . . . . . . . . . . . . . 24 2.2.1 Empirical estimators . . . . . . . . . . . . . . . . . . . 24 2.3 Asymptotic properties of the estimators . . . . . . . . . . . . . 27 2.3.1 Strong consistency . . . . . . . . . . . . . . . . . . . . 27 2.3.2 Asymptotic normality . . . . . . . . . . . . . . . . . . 30 2.4 Markov renewal equation . . . . . . . . . . . . . . . . . . . . . 35 3 Continuous-time semi-Markov process 37 3.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . 37 3.2 Elements of statistical estimation . . . . . . . . . . . . . . . . 41 3.2.1 Empirical estimators . . . . . . . . . . . . . . . . . . . 41 3.2.2 Asymptotic properties of the estimators . . . . . . . . 42 3.3 Markov renewal matrix . . . . . . . . . . . . . . . . . . . . . . 46 4 3.4 Markov renewal equation . . . . . . . . . . . . . . . . . . . . . 49 3.5 Hazard rate function . . . . . . . . . . . . . . . . . . . . . . . 50 4 Applications 53 4.1 Application to asthma control data . . . . . . . . . . . . . . . 53 4.1.1 The SemiMarkov R package . . . . . . . . . . . . . . . 54 4.1.2 Script and concluding remarques . . . . . . . . . . . . 59 4.2 Application to Covid-19 pandemic . . . . . . . . . . . . . . . . 65 4.2.1 Application for Tunisia Coronavirus data . . . . . . . . 65 4.2.2 Application for Algeria Coronavirus data . . . . . . . . 70 Conclusion 86 Appendix 87 Bibliography 90 |
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