In Asymptotic Statistics we study the asymptotic behaviour of (aspects of) statistical procedures. Here “asymptotic” means that we study limiting behaviour as the number of observations tends to infinity. A first important reason for doing this is that in many cases it is very hard, if not impossible to derive for instance exact distributions of test statistics for fixed sample sizes. Asymptotic results are often easier to obtain. These can then be used to construct tests, or confidence regions that approximately have the correct uncertainty level. Similarly, determining estimators or other procedures that are optimal in a specific sense, for instance in the sense of minimal mean squared error or variance, is often not possible if the number of observations is fixed. Using asymptotic results is it however in many cases possible to exhibit procedures that are asymptotically optimal.

In this course we begin by treating the mathematical machinery from probability theory that is necessary to formulate and prove the statements of asymptotic statistics. Important are the various notions of stochastic convergence and their relations, the law of large numbers and the central limit theorem (which the students are assumed to know), the multivariate normal distribution, and the so-called delta method. We will use these tools to study the asymptotic behaviour of statistical procedures.

It is assumed that students have at least successfully completed introductory courses on probability theory and statistics and courses on linear algebra and multivariate calculus. It is highly recommended to follow a course on measure theoretic probability.

### Announcements

• The exam will take place online on January 13. A “Final exam” assignment has been made in the ELO. Please have a look at it for all information and rules about the exam.
• There is a new, corrected version of Chapters 6 and 7. See the link to the notes below.
• Short answers to the retake of 2019-2020 can be found below.
• There will be a final Q&A on January 6, from 10:00-11:00

### Course information

• Lecturer: Harry van Zanten
• TA: Ivan Barta
• Every week 2 hours of recorded lectures + 1 hour online Q&A about the exercises
• Lecture notes: Aad van der Vaart’s lecture notes: can be downloaded here. Additional chapters on minimax lower bounds and high-dimensional models: here.
• Recommended literature: Asymptotic Statistics, by A.W. van der Vaart, Cambridge University Press.
• Exams: single final exam and a retake.

### Online teaching

Due to the Covid crisis the entire course will be online this year. Every week the schedule is as follows:

• Every week recorded lectures will be made available before the scheduled time slot (Wednesdays at 10:00). See the table below for links to the recordings. You will receive a password for the videos via email.
• Starting from September 16th, there is a weekly an online exercise class via Zoom on Wednesdays, 10:00-11:00, discussing the exercises corresponding to the material of the week before (see the table below). You will receive a Zoom link via email.

### Exam

• To be announced

### Retake

• To be announced

### What we have done so far

 Lecture Date Topic Material Exercises Remarks Video links 1 9/9 introduction, convergence slides+notes Sec. 1.1 up to and including Theorem 1.7 1.1, 1.2, 1.3(i), 1.4, 1.10, 1.15 watch hour 1 from minute 13:45 hour 1, hour 2 2 16/9 convergence rest Sec. 1.1, Sec. 1.2 1.11, 1.12, 1.17, 1.28, 1.29, 1.32 1) Skip proof of Prohorov and Helly.2) There is an error in 1.28. Try to correct it! hour 1, hour 2 3 23/9 multivariate normal Sec. 2.1-2.4 2.1, 2.2, 2.5, 2.8, 2.13, 2.16, 2.17 hour 1, hour 2 4 30/9 chi square test and delta method Sec. 2.5, 3.1 2.22, 2.23(i), 3.1, 3.2, 3.3 Skip Theorem 2.10. hour 1, hour 2 5 7/10 delta method Sec. 3.2, 3.3 3.12, 3.18 watch hour 1 from 6:25 hour 1, hour 2 14/10 live Q&A 6 28/10 M-estimators, consistency Chap. 4 up to and including p. 45 + extra material in slides 4.1, 4.5, 4.8, 4.11(i), 4.12(i, ii) There is extra material about Glivenko-Cantelli theorems under bracketing in the slides hour 1, hour 2 7 4/11 M-estimators, asymptotic normality, MLE rest of Chap. 4 4.13, 4.14, 4.16, 4.18, 4.25 watch hour 1 from 23:55 hour 1, hour 2, hour 3 8 11/11 nonparametric estimation Chapter 5 5.1, 5.2, 5.3, 5.6 hour 1, hour 2 18/11 live Q&A Zoom link will be provided 9 25/11 minimax lower bounds 1 sections 6.1, 6.2 6.1 Material from the second set of lecture notes, see above hour 1, hour 2, hour 3 10 2/12 minimax lower bounds 2 rest of chapter 6 6.2-6.7 hour 1, hour 2 11 9/12 high-dimensional models chapter 7 7.3-7.5, 7.7-7.9 last lecture hour 1, hour 2