This seminar introduces the asymptotic theory of nonparametric statistics. A typical problem of nonparametric statistics is the estimation of a function that is assumed to lie in a function class. Examples are Hölder classes and Sobolev balls. The topic of the seminar is the asymptotic theory of optimal estimation in such settings. We study the performance of estimators and prove lower bounds for minimax risks that are available in the model. A main part of the seminar will be different approaches to obtain such lower bounds. The seminar follows the book [1]
Possible presentation topics are:
Local polynomial estimation. Projection estimation. (Chapters 1.6, 1.7.1., 1.7.2., two talks)
Lower bounds on the minimax risk based on two hypotheses (Chapters 2.1- 2.5, two talks)
Lower bounds on the minimax risk based on many hypotheses (Chapters 2.6, two talks)
Fano's lemma, Assouad's lemma, van Tree's inequality (Chapter 2.7, two talks)
Pinsker's theorem (Chapters 3.1-3.4, two talks)
Each participant is expected to give a 60 minutes talk using the blackboard. A handout containing the most important definitions and results as well as short sketches of the proofs should be prepared for the other participants.
Requirements:
The seminar is for advanced Bachelor students and Master students who want to specialize in statistics and are already familiar with the topics typically covered in the lecture Introduction to Probability and Statistics. Knowledge from the lectures Probability Theory I and Statistics I is useful, but not a prerequisite.
Reference:
[1] Alexandre B. Tsybakov Introduction to Nonparametric Statistics, Springer 2009 Link to PDF file