- Thesis:
- Inauguraldissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften der Universität Mannheim
- urn:nbn:de:bsz:180-madoc-419555
- Author:
- Martin Kroll (University of Mannheim)
- Title:
- Concentration inequalities for Poisson point processes with applications to non-parametric statistics
- Supervisor and examiner:
- Martin Schlather (University of Mannheim)
- Second examiner:
- Jan JOHANNES
- Abstract:
- In the first part of this thesis we derive new concentration inequalities for maxima of empirical processes associated with independent but not necessarily identically distributed Poisson point processes. The proofs are based on a careful application of Ledoux’s entropy method. In the second part of the thesis, we show potential applications of the concentration results derived in the first part to non-parametric statistics: we consider intensity estimation for Poisson point processes from direct (Chapter 3) and indirect (Chapter 4) observations and non-parametric Poisson regression (Chapter 5). For all the considered models we develop a minimax theory (upper and lower bounds) under abstract smoothness assumptions on the unknown functional parameter. We study projection estimators in terms of trigonometric basis functions. The performance of these estimators crucially depends on the choice of a dimension parameter. For all our applications, we propose a fully data-driven selection of the dimension parameter based on model selection. The resulting adaptive estimators either attain optimal rates of convergence or are suboptimal only by a logarithmic factor.
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |