# Student Seminar in Statistics: Inference in Some Non-Standard Regression Problems

Autumn semester 2021

## General information

Lecturer Fadoua Balabdaoui Domagoj Ćevid, Yulia Kulagina, Malte Lodschien Mon 16.00-18.00 HG E 21 VVZ

## Course content

Abstract

The course encompasses a review of some non-standard regression models and the statistical properties of estimation methods in such models.

Objective

The students get to discover some less known regression models which either generalize the well-known linear model (for example monotone regression) or violate some of the most fundamental assumptions (as in shuffled or unlinked regression models).

Content

Linear regression is one of the most used models for prediction and hence one of the most understood in statistical literature. However, linearity might be too simplistic to capture the actual relationship between some response and given covariates. Also, there are many real data problems where linearity is plausible but the actual pairing between the observed covariates and responses is completely lost or at partially. In this seminar, we review some of the non-classical regression models and the statistical properties of the estimation methods considered by well-known statisticians and machine learners. This will encompass:

1. Monotone regression
2. Single index model

Literature

In the following is the tentative material that will be read and studied by each pair of students (all the items listed below are available through the ETH electronic library or arXiv). Some of the items might change.

1. Chapter 2 from the book "Nonparametric estimation under shape constraints" by P. Groeneboom and G. Jongbloed, 2014, Cambridge University Press
2. "Estimating a Convex Function in Nonparametric Regression" by M. Birke and H. Dette, 2007, Scandinavian Journal of Statistics, Volume 34, 384-404
3. "Nonparametric shape-restricted regression" by A. Guntuoyina and B. Sen, 2018, Statistical Science, Volume 33, 568-594
4. "Least squares estimation in the monotone single index model" by F. Balabdaoui, C. Durot and H. K. Jankowski, Journal of Bernoulli, 2019, Volume 4B, 3276-3310
5. "Semiparametric Efficiency in Convexity Constrained Single-Index Model" by A. K. Kuchibhotla, R. K. Patra and B. Sen, 2021, JASA (Theory and Methods), 1-15
6. "Sharp thresholds for high dimensional and noisy sparsity recovery using l1-constrained quadratic programming (Lasso)" by M. Wainwright, 2009, IEEE transactions in Information Theory, Volume 55, 1-19
7. "The Isotron Algorithm: High-Dimensional Isotonic Regression" by A. T. Kalai and R. Sastry, 2009, COLT
8. "Forward selection and estimation in high dimensional single index models", by S. Luo and S. Ghosal, 2016, Statistical Methodology, Volume 33, 172-179
9. "Inference In High-dimensional Single-Index Models Under Symmetric Designs" by H. Eftekhari, M. Banerjee, and Y. Ritov, 2021, JMLR, Volume 22, 1-63
10. "Denoising linear models with permuted data" by A. Pananjady, M. Wainwright and T. A. Courtade and , 2017, IEEE International Symposium on Information Theory, 446-450
11. "Unlinked Monotone Regression" by Fadoua Balabdaoui, Charles Doss and Cecile Durot, 2021, JMLR, Volume 22, 1-60
12. "Uncoupled isotonic regression via minimum Wasserstein deconvolution" by P. Rigollet and J. Weed, 2019, Information and Inference, Volume 00, 1-27

Primary target group and Prerequisites

The students need to be confortable with regression models, classical estimation methods (Least squares, Maximum Likelihood estimation...), rates of convergence, asymptotic normality, etc.

## Announcements

13.09.2021
Welcome to the website of the course Student Seminar in Statistics: Inference in Some Non-Standard Regression Problems!

## Course material and schedule

We will study the materials listed in the section Literature.

The registered students will be divided into pairs to work on the papers. Everyone is expected to participate actively during all lectures. Questions and discussions are strongly encouraged in this class!

The presentations should last roughly 2 x 25 minutes, with a 5-10 minute break in between. One of the assistants will meet with you twice before your presentation, to answer questions about the material and to give feedback on your planned presentation. More detailed guidelines for the presentations will be given during the first class. Please also see the FAQ for further details.

Week Topic Slides
Week 1 (27.09.2021) Introductory Lecture

Week 2 (04.10.2021) Group 1: Basic Estimation Problems with Monotonicity Constraints

• Speakers: Hanchen Gu, Jianing Yang
• Assistants: Domagoj
Week 3 (11.10.2021) Group 2: Estimating a Convex Function in Nonparametric Regression

• Speakers: Zihao Zhang, Martin Dominguez
• Assistants: Domagoj
Week 4 (18.10.2021) Group 3: Nonparametric Shape-Restricted Regression

• Speakers: Simone Scandella
• Assistants: Domagoj
Week 5 (25.10.2021) Group 4: Least Squares Estimation in the Monotone Single Index Model

• Speakers: Anna Heck, Dennis Klung
• Assistants: Yulia
Week 6 (01.11.2021) Group 5: Semiparametric Efficiency in Convexity Constrained Single-Index Model

• Speakers: Rebecca Morger, Ramon Stieger
• Assistants: Malte
Week 7 (08.11.2021) Group 6: Sharp Thresholds for High-Dimensional and Noisy Sparsity Recovery Using ℓ1 -Constrained Quadratic Programming (Lasso)

• Speakers: Erik Beyer, Philipp Zhang
• Assistants: Malte
Week 8 (15.11.2021) Group 7: The Isotron Algorithm: High-Dimensional Isotonic Regression

• Speakers: Antony Awad, Alexander Gudjonsson
• Assistants: Yulia
Week 9 (22.11.2021) Group 8: Forward selection and estimation in high dimensional single index models

• Speakers: Pengxi Liu, Jacob Assan
• Assistants: Yulia
Week 10 (29.11.2021) Group 9: Denoising Linear Models with Permuted Data

• Speakers: Genc Kqiku, Carlos Garcia
• Assistants: Yulia
Week 11 (06.12.2021) Group 10: Unlinked Monotone Regression

• Speakers: Lukas Graz, Jiaqi Shao
• Assistants: Yulia
Week 12 (13.12.2021) Group 11: Uncoupled isotonic regression via minimum Wasserstein deconvolution

• Speakers: Hongruyu Chen, Melcior Pijoan
• Assistants: Yulia

## FAQ

1. How long should the presentation be?

The total presentation time is 50 minutes. Each student should present roughly half of the time. We advise you to split the presentation in two parts of about 25 minutes each, with a 5-10 minute break in between. Please make sure to practice so that you don't go over your time! We highly encourage interaction and discussion with the audience, both during and after your talk. If this happens during your talk, this will not be counted as presentation time.

2. Should I use a certain template for my slides?

You can use any template you like. We recommend using one of the ETH presentation templates.

3. How should the presentation be structured?

The main purpose of the presentation is to transmit knowledge to the audience. So, after reading the material, please take a step back and try to put yourself in the shoes of the audience: What do they already know? What would they find most interesting? What would be helpful examples? We will also provide further guidelines for the presentations during the first lecture.

4. Do I need to bring my own laptop to present my slides?

Ideally, yes. If you do not have a laptop, or you do not have a way of connecting to the projector, please let the assistants know in advance.

5. Will my slides be published somewhere?

Yes, all slides will be published on the course website after the presentation. Please make sure to respect copyright. In particular, if you include any images or tables not created by yourself in the presentation, make sure to include the source of the image/table as well.

6. What is the role of the assistants?

The assistant in charge for your group gives you guidance and feedback prior to your presentation. You will have a chance to meet with the assistant twice before your presentation. The first meeting will be on Thursday, 1.5 weeks before your presentation (it will be Thursday by default but it is possible to reschedule the meeting on mutual agreement). The second meeting will typically take place on Thursday, 0.5 week before your presentation (again, rescheduling rule applies).

7. How should I prepare for the meetings with the assistants?

The first meeting: you should read all material in advance, make a list of questions you have, and make a rough plan of what you would like to present (main concepts, main examples, questions you could pose to the audience to create some interaction, R-example that you could integrate, etc). The second meeting: your presentation should be fully prepared and should be sent to the assistants the day before. During the meeting, you will get feedback on your presentation, and you can clarify any remaining issues.

8. Do I have to attend all lectures?

Yes, attendance at all lectures is compulsory. If you have to miss a class (due to illness or some force-major), please contact Prof. Dr. Fadoua Balabdaoui directly.