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Measure theory and asymptotic statistics (TI1708) Tinbergen Institute 2020-2021 AimTo make students familiar with the mathematical fundamentals of measure theory and asymptotic methods in statistics. This is a crash course, highlighting the main principles, not an in depth treatment of the theory.ContentsPart I: Sigma-algebras, measure, integration w.r.t. a measure, limit theorems, product measure and integration, change of measure, conditional expectation. Part II: Multivariate central limit theorem, quadratic forms, delta-method, moment estimators, Z-and M-estimators, consistency and asymptotic normality, maximum likelihood estimators.LiteratureThe first weeks of the course are mainly based on the condensed lecture notes Measure theoretic probability. [The extended version of these lecture notes contains a lot of additional material that will not be covered in this course. For this part also the first chapters in Steve Shreve (2004), Stochastic Calculus for Finance II, Continuous-Time Models can be useful. This book is written with a wide range of applications in Mathematical Finance in mind, but will not be used.]The second part of the course will be based on the lecture notes Mathematische Statistiek by A.W. van der Vaart (title and preface are in Dutch, content in English). [These lecture notes have been considerably expanded, resulting in the (advanced) textbook Asymptotic Statistics by the same author. This book will not be used in the course.] See also the extra notes that contain some additional material, especially for the parts on Asymptotic statistics. ExaminationWe will follow the usual conventions for TI core courses, i.e. there will be a written closed book exam and homework assignments. Homework has to be handed every week to the TAs (due dates will be communicated by them). Your final grade F will be a weighted average of your result H of the homework assignments and the result E of the written exam: F=0.85*E+0.15*H. Homework assignments may be made in pairs (at most TWO people). The exercises at the exam will be at the level of the homework and tutorial sessions, but may also contain some theory. You don't have to know all proofs by heart, but at least the gist of them. Important theorems and definitions you are required to know. The written exam is on October 21, 2020, 11:00-14:00 in room 1.61 (TIA) and online. During the exam you are allowed to use printed copies of the two sets of lecture notes.This course is for a major part on new topics since 2017. There are a number of old exam questions available, some of them from the time that Measure theory was part of an older course with different other topics. From a message by Andreas Pick: "We will get a large room such that the students can write their exams with sufficient social distancing. Students can request to write exams remotely and, in this case, we will organise remote proctoring." PeopleLectures by Peter Spreij, teaching assistance by Xu Lin and Diego Dabed.ScheduleDue to the presence of the corona virus, the lectures will be online. Lectures mostly on Wednesdays, first lecture on 2 September 2020; recordings will be made available before the scheduled time, or streamed (to be decided). The first lecture will be streamed and recorded. There will be an extra lecture on Friday September 25, 2020 (details follow). TA sessions on Mondays in two groups, 9:30-10:45 and 11:00-12:15. First session on September 7, 2020.LocationTinbergen Institute Amsterdam, Gustav Mahlerplein 117, 1082 MS Amsterdam
Programme |
| 1 |
Class: From Spreij, Sections 1.1, 1.2, 1.3, (most of) 1.4, Most of 2.1 (ignore everything on topology), 2.2 (skip the proof of Theorem 2.10).
Tutorial: Make Exercises 1.6, 1.9, 2.2, 2.4. Homework: Make Exercises 1.1, 1.2, 1.4, 2.1; Read Section 1.5, Proposition 2.12 with the proof, the statement of Theorem 2.6 (it is instructive to study the proof as well). Note: On September 3, I uploaded a revised version of the lecture notes. Main changes are a few lines below Definition 2.11 and additional Exercises 1.11, 1.12, 2.12 (try them, they should not be difficult; or just have a look). |
| 2 |
Class: Most of Sections 3.1 (some tedious proofs will be skipped), 3.2, 3.4, 3.5 and 3.6 (sketchy), 4.1 up to Theorem 4.5, very brief mentioning of Section 4.2 with emphasis on relation between independence and product measures.
Tutorial: Make Exercises 3.3, 3.4, 3.8, 4.4. Homework: Make Exercises 3.9, 3.11, 4.5, 4.6. Pay some (superficial) attention to parts (proofs!) that I have skipped. Note: Some of the extra material (a few lines only) in below Definition 2.11 has now (September 5) been moved to Section 4.2. There are now also some extra notes on norms and metrics. You may want to have a quick look. Metrics will play a role in Week 4. |
| 3 |
Class: Quick reminder of some essentials in Section 3.3, connection to Section 5.1 (Proposition 5.3), Section 5.2; Section 6.1, most of Section 6.2 (in Theorem 6.7 emphasis on monotone convergence, parts of Theorem 6.8 with their proofs), very superficial treatment of Section 6.3.
Tutorial: Make Exercises 5.1, 5.4, 6.1, 6.6. Homework: Make Exercises 5.3 (optional), 5.5, 6.3. |
| 4 |
Class: Chapter 1 of Van der Vaart, and most parts of Sections 2.1 and 2.2. Chapter 6 (Appendix) is supposed to be known; it is mainly a collection of results that you have seen in the previous weeks, possible in different notation.
Tutorial: Chapter 1: Exercises 1, 6, 7, 15, 21. Homework: Chapter 1: Exercises 12, 22, 25; and optional: 19, 20a. |
| 5 |
Class: Sections 2.3, 2.4, first part of Section 2.5; Sections 3.1, 3.2.
Tutorial: Chapter 2: Exercises 9+10, 11; Chapter 3: 4, 20. Homework: Chapter 2: Exercises 3, 4 (hint: what is the distribution of $z^T X$?); Chapter 3: 1, 11, 12. Look also at the extra notes for the recordings of Week 6. |
| 6 |
Class: Sections 3.3, 4 and 4.1.
Tutorial: Chapter 4: Exercises 1, 2, 5. Homework: Chapter 4: Exercises 3, 6, 7. |
| 7 |
Class: Sections 4.2 (but not Subsection 4.2.1) and 4.3.
Tutorial: Chapter 4: Exercises 4, 12, 13, 21. Homework (last week, so optional!): Chapter 4: Exercises 10, 11(i,ii), 19, 23. |