Statistical distances for model validation and clustering. Applications to flow cytometry and fair learning.

Abstract

This thesis has been developed at the University of Valladolid and IMUVA within the framework of the project Sampling, trimming, and probabilistic metric techniques. Statis- tical applications whose main researchers are Carlos Matr an Bea and Eustasio del Barrio Tellado. Among the lines of research associated with the project are: model validation, Wasserstein distances and robust cluster analysis. It is precisely the work carried out in these elds that gives rise to chapters 1,2 and 4 of this report. The work done in the eld of fair learning with Professor Jean-Michel Loubes, frequent collaborator with Valladolid's team, during the international stay at the Paul Sabatier University of Toulouse, is the basis of Chapter 3 of this report. Therefore, this thesis is an exposition of the problems and results obtained in the di erent elds previously mentioned. Due to the diversity of topics, we have decided to base chapters on the works published or submitted to the present date, and therefore each chapter has a structure relatively independent of the others. In this way Chapter 1 is based on the works [del Barrio et al., 2019e,del Barrio et al., 2019d], Chapter 2 is based on the work [del Barrio et al., 2019c], Chapter 3 on the work [del Barrio et al., 2019b] and Chapter 4 shows results of a work in progress. In this introduction our objective is to present the main challenges we have faced, as well as to brie y present our most relevant results. On the other hand, each chapter will have its own introduction where we will delve into the topics discussed below. With this in mind, our intention is that the reader will have a general idea of what he or she will nd in each chapter and in this way will have the necessary information to face the more technical discussions that will be found there. Due to the diversity of topics dealt with in this report, we propose a non-linear reading. We suggest that the reader, after reading a section of the Introduction, moves to the corresponding chapter. In this way the reader will have the relevant information more at hand and will be able to follow better the exposition in each chapter. If on the other hand there is a sequential reading of the document, we apologize in advance for some repetitions and reiterations, which nevertheless seem to us to contribute positively to the understanding of this work.