Statistics 2013-2014
M.Phil. course Tinbergen Institute

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Contents

The course is intended for students who have a deficiency in probability and statistics. It starts off with the very first principles of probability and quickly passes on to essential statistical techniques. Estimation and testing theory will be reviewed, including maximum likelihood estimators, likelihood ratio test and (least squares) regression. The course is based on John A. Rice, Mathematical Statistics and Data Analysis, Duxbury Press, Belmont, California. From this book we will treat a good deal of the chapters 2-6, 8, 9 and 14. All together the topics will be treated in 7 lectures. Students are required to study the corresponding theory and examples in the book as well as to make accompanying exercises.

In the course we treat the following topics.

Sample spaces, probability measures, distribution functions, random variables with discrete and continuous distributions, functions of random variables, multivariate distributions, random vectors, independent random variables, conditional distributions, functions of random vectors and their distributions, expectation and variance, covariance and correlation, the law of large numbers, central limit theorem, chi-square and t-distributions, estimation, method of moments, maximum likelihood, large sample theory, confidence intervals, Cramer-Rao bound, hypothesis testing, Neyman-Pearson paradigm, likelihood ratio tests, confidence intervals, linear regression, least squares estimation of regression parameters, testing regression hypotheses.

Literature

John A. Rice, Mathematical Statistics and Data Analysis, 2nd Edition, Duxbury Press (1995, ISBN: 053420934-3), or 3rd Edition (2007, ISBN: 0534399428). Both editions can be used for this year's course, the new edition has more examples (also from financial statistics) and exercises. For the third edition there is a list of errata. Second hand copies of the book are sometimes available at Amazon Germany. Browse the web for other offers.

You may also want to see pdf copies of some slides, or the extra notes complementing some of the material in the book.

Prerequisites

Some knowledge of elementary mathematics. The course Mathematics 1 by Jan Brinkhuis offers more than enough. Basic knowledge of probability is required up to the level of chapter 1 of Rice. This chapter will NOT be treated in the course and students are supposed to be familiar with its contents. Chapter 2 will not be treated in detail, only highlights. Students should study the many examples of distributions themselves.

People

Peter Spreij (lecturer), Hao fang (teaching assistant)

Locations and Schedule

First lecture on Wednesday October 30, 13:30-16:15, other lectures on Wednesday afternoons starting November 6, 14:00-16:45; last lecture on December 11. TA sessions on Wednesdays: 12:30-13:30, with the exception of the last session on December ??.

Examination

Homework assignments and written exam. Homework has to be handed every week on a Wednesday, see the schedule below. During the written exam you are allowed to use the book and a pocket calculator. 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=0.8*E+0.2*H, provided E >= 4.4. If E < 4.4, then F=E.
Date and time of the written exam: December 18, 13:30-16:30 at TIA.
As an example of what could be asked, you could have a look at the exams of December 21, 2005 and June 30, 2006.

The final results of this year (the F above) will be presented here after the grading of the written exam.


Programme

The schedule below might see some small changes during the course. Check this page regularly for updates!! You should the schedule as follows. "Week x" contains the content of the class that week, the exercises on the corresponding theory, of which a part will be the programme of the TA session, usually in week x+1, and the resulting homework to be handed at the same TA session. At the end of the course, this scheme will be compressed a bit. The exercises listed below are all useful, but those marked with an asterisk (*) deserve special attention. The numbering of the exercises corresponds to the 2nd edition of the book. The numbering of the exercises in the 3rd edition deviates from the numbering in the 2nd edition. We have a conversion table that lists the correspondence between the two editions. It also turned out that the section numbering (sometimes) and page numbering has been changed between the two editions, see the table below.

2nd EDITION3rd EDITION
pages 202-203pages 216-218
section 8.6section 8.7
sections 9.1-9.3sections 9.1-9.2
section 9.4section 9.3
section 9.5section 9.4

FROM WEEK TO WEEK
1 Rice, chapter 2 and 3 (main themes only); students should study the many examples of distributions themselves, but skip parts of chapter 3 that require more than basic knowledge of multiple integrals.
Exercises: chapter 2: 5, 23, 33, 41, 44*, 53, 55*, 59*; chapter 3: 7, 17(a,b), 32(a), 34*, 37*, 38
Homework: none
2 Rice, sections 4.1-4.3 (except Markov and Chebychev inequalities)
Exercises: chapter 4: 2, 4, 6, 12, 31, 34*, 45*, 46*, 53*, 71(a)
Homework: chapter 4: 32, 45 (due November 13)
3 Rice, chapter 5 (skip the considerations involving moment generating functions), sections 6.1, 6.2, 6.3, extra on multivariate normal distributions (see slides)
Exercises: chapter 5: 1*, 3*, 9, 12*, 13, 15, 17*, 23, 26
Homework: chapter 5: 16, chapter 6: 9 (due November 20)
4 Rice, sections 8.3-8.5.2 (Theorem B next week)
Exercises (November 8): chapter 8: 4*, 5*, 8, 17* and (in the numbering of the 3rd!! edition) 5abc, 12 (ignore all questions on Fisher information and on sufficient statistics)
Homework: chapter 8: 14abc, 19ab (due November 27)
5 Rice, sections 8.5.2 (continued) 8.5.3 (you also read pages 202, 203 in the 2nd edition, pages 217, 218 in the 3rd edition), 8.6, and a small part of 9.2
Exercises: chapter 8: 39abc*, 42, 44abc, 49, 52*
Homework: chapter 8: 17, 33 (NEW: the bootstrap exercise should be replaced with 8:33 in the 3rd edition); chapter 9: 2 (due December 4) and read section 9.2 until the Neyman-Pearson lemma
6 Rice, sections 9.2, 9.3, 9.5 (very briefly)
Exercises: chapter 9: 1, 3*, 5, 7*, 9; read the section on the generalized likelihood ratio test. If you come across p-values, just skip it. It will be on the agenda for the last week as well as how one can use confidence intervals for testing.
Homework: chapter 9: 11, 16 (due December 11)
7 Rice, p-value, sections 9.4, 14.1-14.4 (emphasis on 14.3, 14.4)
Exercises (might be changed): chapter 9: 15*, chapter 14: 3*, 4, 8*, 9, 12*, 15
Homework: chapter 14: 5, 16, 25* (due December 18)


To the Korteweg-de Vries Institute for Mathematics or to the homepage of Peter Spreij.