Seminar in Statistics: Statistical Inference under Shape Restrictions
Spring semester 2017
General information
Lecturer  

Assistants  Emilija Perkovic, Marco Eigenmann 
Lectures  Mon 1517 HG G 26.5 >> 
Course catalogue data  >> 
Course content
AbstractStatistical 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, kmonotonicity or logconcavit.
ObjectiveThe 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 socalled 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.
Announcements

January 26th, 2017:
Beginning of lecture and assignment of the topics: Monday, 20/02/2017. 
January 26th, 2017:
First presentation: Monday, 27/02/2017.
Course materials
Week  Topic  Material  Students' Slides 

Week 1 (20/02/2017)  Introduction / Assignment of the topics.  
Week 2 (27/02/2017)  Topic 1: Isotonic Regression



Week 3 (06/03/2017)  Topic 2: Generalized isotonic regression problems



Week 4 (13/03/2017)  Topic 3: Testing the equality of ordered means: Likelihood ratio test in the Normal case



Week 5 (20/03/2017)  Topic 4: More on monotone problems: Estimation of a monotone density and a distribution function from Current Status data



Week 6 (27/03/2017)  Topic 5: Algorithms and computation in shapeconstrained problems



Week 7 (03/04/2017)  Topic 6: Estimation of a convex ROC curve



Week 8 (10/04/2017)  Topic 7: Mixtures of Exponential distributions



Week 9 (08/05/2017)  Topic 8: Shape restricted nonparametric regression using Bernstein polynomials



Week 10 (15/05/2017)  Topic 9: Maximum Likelihood estimation of a logconcave density and its distribution function: Basic properties and uniform consistency



Week 11 (22/05/2017)  Topic 10: On asymptotics of the discrete convex LSE of a probability mass function



Week 12 (29/05/2017)  Topic 11: Estimation of a discrete probability under constraint of kmonotonicity

