Skip to Main content Skip to Navigation
Theses

Contributions à l’estimation à noyau de fonctionnelles de la fonction de répartition avec applications en sciences économiques et de gestion

Abstract : The income distribution of a population, the distribution of failure times of a system and the evolution of the surplus in with-profit policies - studied in economics and management - are related to continuous functions belonging to the class of functionals of the distribution function. Our thesis covers the kernel estimation of some functionals of the distribution function with applications in economics and management. In the first chapter, we offer local polynomial estimators in the i.i.d. case of two functionals of the distribution function, written LF and TF , which are useful to produce the smooth estimators of the Lorenz curve and the scaled total time on test transform. The estimation method is described in Abdous, Berlinet and Hengartner (2003) and we prove the good asymptotic behavior of the local polynomial estimators. Until now, Gastwirth (1972) and Barlow and Campo (1975) have defined continuous piecewise estimators of the Lorenz curve and the scaled total time on test transform, which do not respect the continuity of the original curves. Illustrations on simulated and real data are given. The second chapter is intended to provide smooth estimators in the i.i.d. case of the derivatives of the two functionals of the distribution function presented in the last chapter. Apart from the estimation of the first derivative of the function TF with a smooth estimation of the distribution function, the estimation method is the local polynomial approximation of functionals of the distribution function detailed in Berlinet and Thomas-Agnan (2004). Various types of convergence and asymptotic normality are obtained, including the probability density function and its derivatives. Simulations appear and are discussed. The starting point of the third chapter is the Parzen-Rosenblatt estimator (Rosenblatt (1956), Parzen (1964)) of the probability density function. We first improve the bias of this estimator and its derivatives by using higher order kernels (Berlinet (1993)). Then we find the modified conditions for the asymptotic normality of these estimators. Finally, we build a method to remove boundary effects of the estimators of the probability density function and its derivatives, thanks to higher order derivatives. We are interested, in this final chapter, in the hazard rate function which, unlike the two functionals of the distribution function explored in the first chapter, is not a fraction of two linear functionals of the distribution function. In the i.i.d. case, kernel estimators of the hazard rate and its derivatives are produced from the kernel estimators of the probability density function and its derivatives. The asymptotic normality of the first estimators is logically obtained from the second ones. Then, we are placed in the multiplicative intensity model, a more general framework including censored and dependent data. We complete the described method in Ramlau-Hansen (1983) to obtain good asymptotic properties of the estimators of the hazard rate and its derivatives and we try to adopt the local polynomial approximation in this context. The surplus rate in with-profit policies will be nonparametrically estimated as its mathematical expression depends on transition rates (hazard rates from one state to another) in a Markov chain (Ramlau-Hansen (1991), Norberg (1999))
Document type :
Theses
Complete list of metadatas

Cited literature [165 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-01691084
Contributor : Abes Star :  Contact
Submitted on : Tuesday, January 23, 2018 - 4:06:07 PM
Last modification on : Saturday, December 7, 2019 - 3:50:01 AM
Long-term archiving on: : Thursday, May 24, 2018 - 4:04:57 PM

File

TH2017MadaniSoffana.pdf
Version validated by the jury (STAR)

Identifiers

  • HAL Id : tel-01691084, version 1

Collections

Citation

Soffana Madani. Contributions à l’estimation à noyau de fonctionnelles de la fonction de répartition avec applications en sciences économiques et de gestion. Gestion et management. Université de Lyon, 2017. Français. ⟨NNT : 2017LYSE1183⟩. ⟨tel-01691084⟩

Share

Metrics

Record views

470

Files downloads

848