Measure theory and asymptotic statistics 2017-2018

Measure theory and asymptotic statistics
2017-2018 (TI1708)
Tinbergen Institute

Aim

To 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.

Part 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.

Literature

The first weeks of the course are mainly based on Steve Shreve (2004), Stochastic Calculus for Finance II, Continuous-Time Models, Springer. There are many other texts on measure theory. For free are available the lecture notes Measure theoretic probability. These notes are used for courses in the master programme for mathematics students Stochastics and Financial Mathematics. The second part of the course will be based on the lecture notes by A.W. van der Vaart (title and preface are in Dutch, content in English).

Examination

We 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 Jan Meppe. 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. 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 23, 2017, 13:30-16:30. During the exam you are allowed to use printed copies of the book by Shreve and the lecture notes by Van der Vaart.
This course is for a major part on new topics. Measure theory was also part of an older course, and on this topic there are a number of old exam questions available.

People

Lectures by Peter Spreij, teaching assistance by Jan Meppe.

Schedule

Fall semester, 1st half. First lecture on Wednesday 6 September, 09:00-11:45. All next lectures, from 8 September on, on Fridays, 13:45-16:30 in room 1.01, except on October 6 in room 1.60. Tutorials after the lectures.

Location

Tinbergen Institute Amsterdam, Gustav Mahlerplein 117, 1082 MS Amsterdam

Programme
(please, look out for updates; )

1
Class: From Shreve, most of Sections 1.1, 1.2, 1.3
Homework : Read before the second class in the book the sections that I have treated in the classroom, including those pieces that have been skipped. In particular pay attention to the Lebesgue integrals following Definition 1.3.7 and observe the similarities and differences with expectations. Take notice of (the gist of what is written in) Sections A.1 and read also section A.2.
Tutorial (combined with tutorial 2): Exercise 1.5 and Exercises 2, 3 from the additional exercises,
Homework (combined with homework 2): Read the Sections that have been treated in the second lecture, also the parts and examples that have been skipped. Read also pages 49, 50 as an introduction to the next class. Make Exercises 1.3 of Section 1.9.

2
Class: Most of Sections 1.4, 1.5, 1.6 and Appendix B
Tutorial (combined with tutorial 1): Exercise 1.15, additional Exercises 5, 7 (if there is enough time, otherwise drop one exercise)
Homework (combined with homework 1): 1.9, 1.11, additional Exercise 8.

3
Class: Most of Sections 2.2, 2.3 (Definitions 2.3.5 and 2.3.6 will be treated next time, perhaps).
Tutorial: Exercises 2.2, 2.10 and 2.4 if there is time left.
Homework: Make Exercises 2.1, 2.6, 2.7, 2.9 and read the Summary Sections 1.7 and 2.4. Also read the examples in Shreve, and pay special attention to the first three pages of Section 2.3, but ignore Equations (2.3.1) - (2.3.3) and the surrounding text.

4
Class: Chapter 1 of the lecture notes, and most parts of Sections 2.1 and 2.2. Chapter 6 (Appendix) is supposed to be known.
Tutorial: Chapter 1: Exercises 1, 6, 7, 15, 21.
Homework: Chapter 1: Exercises 12, 19, 20a, 22, 25.

5
Class: Sections 2.3, 2.4, first part of Section 2.5; Sections 3.1, 3.3, perhaps Section 3.2.
Tutorial: Chapter 2: Exercises 9+10, 11; Chapter 3: 4, 20.
Homework: Chapter 2: Exercises 3, 4; Chapter 3: 1, 11, 12.

6
Class: Sections 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 (optional!): Chapter 4: Exercises 10, 11(i,ii), 19, 23.

Links

Korteweg-de Vries Institute for Mathematics
Master Stochastics and Financial Mathematics

1	Class: From Shreve, most of Sections 1.1, 1.2, 1.3 Homework : Read before the second class in the book the sections that I have treated in the classroom, including those pieces that have been skipped. In particular pay attention to the Lebesgue integrals following Definition 1.3.7 and observe the similarities and differences with expectations. Take notice of (the gist of what is written in) Sections A.1 and read also section A.2. Tutorial (combined with tutorial 2): Exercise 1.5 and Exercises 2, 3 from the additional exercises, Homework (combined with homework 2): Read the Sections that have been treated in the second lecture, also the parts and examples that have been skipped. Read also pages 49, 50 as an introduction to the next class. Make Exercises 1.3 of Section 1.9.
2	Class: Most of Sections 1.4, 1.5, 1.6 and Appendix B Tutorial (combined with tutorial 1): Exercise 1.15, additional Exercises 5, 7 (if there is enough time, otherwise drop one exercise) Homework (combined with homework 1): 1.9, 1.11, additional Exercise 8.
3	Class: Most of Sections 2.2, 2.3 (Definitions 2.3.5 and 2.3.6 will be treated next time, perhaps). Tutorial: Exercises 2.2, 2.10 and 2.4 if there is time left. Homework: Make Exercises 2.1, 2.6, 2.7, 2.9 and read the Summary Sections 1.7 and 2.4. Also read the examples in Shreve, and pay special attention to the first three pages of Section 2.3, but ignore Equations (2.3.1) - (2.3.3) and the surrounding text.
4	Class: Chapter 1 of the lecture notes, and most parts of Sections 2.1 and 2.2. Chapter 6 (Appendix) is supposed to be known. Tutorial: Chapter 1: Exercises 1, 6, 7, 15, 21. Homework: Chapter 1: Exercises 12, 19, 20a, 22, 25.
5	Class: Sections 2.3, 2.4, first part of Section 2.5; Sections 3.1, 3.3, perhaps Section 3.2. Tutorial: Chapter 2: Exercises 9+10, 11; Chapter 3: 4, 20. Homework: Chapter 2: Exercises 3, 4; Chapter 3: 1, 11, 12.
6	Class: Sections 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 (optional!): Chapter 4: Exercises 10, 11(i,ii), 19, 23.

Aim

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