Estimation and Testing under Sparsity

Lecturer: Prof. Sara van de Geer

Content: In high-dimensional models the number of parameters p is larger than the number of observations n. Therefore, classical (asymptotic) theory needs new methods and paradigms for estimation and testing. One of the key concepts here is "sparsity" which says that most of the parameters are actually not relevant and can be set to zero. A popular way to take sparsity into account is regularizing using the l_1-penalty. This leads to two lines of research. Firstly, we need to study the statistical properties of l_1-regularized estimators and related issues, for example their role as initial estimators in a one-step procedure for the construction of asymptotically linear estimators. Secondly, the l_1-approach has a special geometry which one can study in terms of properties of empirical processes. Therefore the lectures have two intertwined parts: one where statistical theory plays the main role and a second where probability theory is studied. Most results presented will be given a full proof, perhaps with parts left as exercises.

Lecture:
Mo 08-10 HG G 26.1

Material: Script