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|All Authors / Contributors:||
Nicolas Verzelen; Pascal Massart; Université de Paris-Sud. Faculté des Sciences d'Orsay (Essonne).
|Notes:||Thèse soutenue en co-tutelle.
Thèse rédigée entièrement en anglais.
|Description:||1 vol. (233 p.) : ill. en noir et en coul. ; 30 cm.|
|Responsibility:||Nicolas Verzelen ; [sous la direction de] Pascal Massart.|
This work is linked to the theories of non-parametric statistics, statistical learning, and spatial statistics. Its goal is to provide and study statistical procedures for Gaussian graphical models. Graphical models have emerged as useful tools for modelling complex systems in many fields such as genomics or spatial analysis. The recent availability of a huge amount of data challenges us with new issues: the number of variables under study is possibly much larger than the sample size. This motivates the search for methods that remain valid in a high-dimensional setting. In this setting, three main issues are considered: the goodness of fit test of the graph of a Gaussian graphical model, the graph estimation of a Gaussian graphical model, and the covariance estimation of a Gaussian graphical model or more generally of a Gaussian vector. Furthermore, we use graphical models to study the covariance estimation of a stationary Gaussian field on a lattice. Our approach is based on the connection between Gaussian graphical model and linear regression with Gaussian design. This connection motivates the use of model selection techniques by penalization. The procedures introduced to analyze each of the four previous issues satisfy non-asymptotic oracle inequalities and are adaptive in the minimax sense. All these results still hold in a high-dimensional setting. The practical efficiency of the procedures is assessed on simulated and real-world data.