Student Seminar in Statistics: Nonparametric Estimation under Shape-Constraints

General information

Course content

Statistical inference based on a random sample can be performed under additional shape restrictions on the unknown entity to be estimated (regression curve, probability density,...). Under shape restrictions, we mean a variety of constraints. Examples thereof include monotonicity, bounded variation, convexity, k-monotonicity or log-concavity.

The main goal of this Student Seminar is to get acquainted with the existing approaches in shape constrained estimation. The students will get to learn that specific estimation techniques can be used under shape restrictions to obtain better estimators, especially for small/moderate sample sizes. Students will also have the opportunity to learn that one of the main merits of shape constrained inference is to avoid choosing some arbitrary tuning parameter as it is the case with bandwidth selection in kernel estimation methods. Furthemore, students will get to read about some efficient algorithms that can be used to fastly compute the obtained estimators. One of the famous algoritms is the so-called PAVA (Pool Adjacent Violators Algorithm) used under monotonicity to compute a regression curve or a probability density. During the Seminar, the students will have to study some selected chapters from the book "Statistical Inference under Order Restrictions" by Barlow, Bartholomew, Bremner and Brunk as well as some "famous" articles on the subject.