Measure theory and asymptotic statistics
2019-2020 (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.

Contents

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 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.

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 the ASSISTANT. 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 23??, 13:30-16:30 ??. 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.

People

Lectures by Peter Spreij, teaching assistance by Timo Schenk.

Schedule

Fall semester, 1st half. Lectures mostly on Wednesdays, 13:30-16:15; first lecture on 4 September 2018, 13:30-16:15; fourth lecture on Thursday 19 September, 09:30-12:15, following the third lecture on Wednesday 18 September. Tutorials are on Monday morning, 11:00-12:30.

Location

Tinbergen Institute Amsterdam, Gustav Mahlerplein 117, 1082 MS Amsterdam

Programme
(please, look out for updates; )

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), no time for Section 2.3.
Tutorial: Make Exercises 1.6, 1.9, 2.2, 2.4.
Homework: Make Exercises 1.1, 1.2, 1.4, 2.1 (due Friday September 13, 12:00); Read Section 1.5, Proposition 2.12 with the proof and the statement of Theorem 2.6, and Definition 2.11 and Corollary 2.13 from Section 2.3 (skip the proofs).
2
Class: Most of Sections 3.1 (some tedious proofs will be skipped), 3.2, 3.4 (without Propositions 3.29, 3.30), 3.5 (up to Proposition 3.34), 4.1 up to Theorem 4.5, very brief mentioning of Section 4.2.
Tutorial: Make Exercises 3.3, 3.4, 3.9, 4.4.
Homework: Make Exercises 3.10, 3.12, 4.5, 4.6 (due September 20). Have a glance at the main issues of Section 4.2.
3
Class: Section 5.1: mainly Proposition 5.3 only, Section 5.2; Section 6.1, most of Section 6.2 without Theorem 6.7 (not all proofs). General idea 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 (due September 27).
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.
Tutorial: Chapter 1: Exercises 1, 6, 7, 15, 21.
Homework: Chapter 1: Exercises 12, 25; and optional: 19, 20a, 22 (due September 27).
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 (due October 4).
6
Class: Sections 4 and 4.1.
Tutorial: Chapter 4: Exercises 1, 2, 5.
Homework: Chapter 4: Exercises 3, 6, 7 (due October 11).
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 (due October 18).





Links

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