Contributions to distribution function kernel estimation with applications to economics and management.

Summary The distribution of incomes of a population, the distribution of failure times of a material and the evolution of profits of life insurance contracts - studied in economics and management - are related to continuous functions belonging to the class of distribution functionals. Our thesis deals with the kernel estimation of distribution functionals with applications in economics and management sciences. In the first chapter, we propose local polynomial estimators in the i.i.d. framework of two distribution functionals, denoted LF and TF , useful to produce 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 local polynomial estimators. Until then, Gastwirth (1972) and Barlow and Campo (1975) had defined piecewise continuous estimators of the Lorenz curve and the total normalized test time, which did not respect the continuity property of the initial curves. Illustrations on simulated and real data are proposed. The second chapter aims at providing local polynomial estimators in the i.i.d. framework of the successive derivatives of the distribution functionals explored in the previous chapter. Apart from the estimation of the first derivative of the TF function, which is treated using the smooth estimation of the distribution function, the estimation method employed is the local polynomial approximation of the distribution functionals detailed in Berlinet and Thomas-Agnan (2004). Various types of convergence as well as asymptotic normality are obtained, including for the density and its successive derivatives. Simulations appear and are commented. The starting point of the third chapter is the Parzen-Rosenblatt estimator (Rosenblatt (1956), Parzen (1964)) of the density. We first improve the bias of the Parzen-Rosenblatt estimator and its successive derivatives using higher order kernels (Berlinet (1993)). We then prove the new asymptotic normality conditions of these estimators. Finally, we construct an edge effect correction method for estimators of density derivatives using higher order derivatives. The last chapter focuses on the hazard rate, which unlike the two distribution functionals treated in the first chapter, is not a ratio of two linear distribution functionals. In the i.i.d. framework, the kernel estimators of the hazard rate and its successive derivatives are constructed from the kernel estimators of the density and its successive derivatives. The asymptotic normality of the first estimators is logically obtained from that of the second. We then place ourselves in the multiplicative intensity model, a more general framework encompassing censored and dependent data. We conduct the Ramlau-Hansen (1983) forward procedure to obtain good asymptotic properties of the estimators of the hazard rate and its successive derivatives and then attempt to apply the local polynomial approximation in this context. The rate of accumulation of surplus in the area of profit sharing can then be estimated nonparametrically since it depends on the transition rates (hazard rate from one state to another) of a Markov chain (Ramlau-Hansen (1991), Norberg (1999)).