Abstract
In this thesis, we focus on the statistical comparison tests of two samples based on ranks on the one hand and the cluster detection method based on spatial scan statistics on the other hand. In both cases, the work was performed using functional data. The goal is to extend the methods developed in the univariate case, i.e. for R-valued random variables to the functional case, i.e. by using random variables valued in functional space. In the first part of this thesis, we study the median test based on ranks in the univariate case. We propose an extension of this latter for functional data. Then, we study the asymptotic behavior of its statistic under the null hypothesis. This extension is compared to other existing parametric and nonparametric statistics using simulated and real data to study its performance. In the second part, we introduce a nonparametric spatial scan statistic for functional data. It is derived from the Wilcoxon-Mann-Whitney statistic defined in an Hilbert space. The proposed scan method is applied on simulated data to study its performance, then on real data to extract characteristics of the demographic evolution of the Spanish population. In the last part, we develop an R package called HDSpatialScan. It allows to apply spatial scan statistics developed recently for functional data including the nonparametric scan statistic that we developed in this thesis. This package facilitates the use of the scanning methods and allows to plot the detected clusters with a simple and fast manner.