- Thesis:
- Master in Mathematics
- Author:
- Marlena Weidenauer
- Title:
- Estimation of non-smooth functionals under sparsity
- Supervisors:
- Jan JOHANNES
- Abstract:
- When analysing high-dimensional data, one often encounters sparsity, a phenomenon where not all parameters of a model have a significant influence on the data. Models incorporating the assumption of sparsity are therefore of particular interest. In this thesis, we specifically treat sparsity as the case where only a small fraction of entries of a parameter vector differs from zero. We study the problem of estimating a family of non-smooth functionals, including the l1-norm, of sparse normal means modelled in a Gaussian sequence model. We determine non-asymptotic minimax optimal rates of estimation on the classes of sparse vectors, where we distinguish between two regimes: the sparse and the dense zone and construct two estimators, one for each regime, that achieve these rates. To establish lower bounds on the mini max risk, we apply a reduction scheme that generalizes the classical approach based on two hypotheses. The theoretical results are complemented by a simulation study.
- Reference:
- O. Collier, L. Comminges, and A. B. Tsybakov. On estimation of nonsmooth functionals of sparse normal means, Bernoulli 26:3,1989–2020, 2020.
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |