Statistics (TI1707)
Tinbergen Institute 2020-2021

### 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.

Most important are the slides of the first lecture; this lecture will only be presented online. Take notice of these slides before the first lecture on location on September 2, 2020, and preferably also before the first tutorial session on September 1, 2020.
You may also want to see pdf copies of some slides used in the lectures, or the extra notes complementing some of the material in the book.

### Prerequisites

Some knowledge of elementary mathematics. The course Mathematics 1 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), Aishameriane Schmidt and Saeed Badri (teaching assistants)

### Locations and Schedule

In spite of the presence of the corona virus, lectures are planned to be on location (TI Amsterdam). Lectures on location on Wednesdays, 11:00-14:00, starting on 2 September 2020. However, the first lecture will be available online only, a link will be sent by mail. You are supposed to know the contents of the online lecture before the start of the lectures on location. TA sessions on Tuesdays; in two groups, 15:30-16:45 and 17:00-18:15. First TA sessions on September 1, last sessions TBA. Students need to check with TAs for the spreadsheet (communicated by mail) with the session they will attend.

### Examination

Homework assignments and written exam. Homework has to be handed every week on the due dates determined by the TAs. 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=0.85*E+0.15*H. Date and time of the written exam: October 20, 2020, 11:00-14:00 in room 1.61 (TIA) and online. As an example of what could be asked, you could have a look at the collection of exam questions.
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."

### Interactive website

You may be interested in this interactive website where you can experiment yourself with various topics in probability and statistics. The website accompanies the textbook Statistics: The Art and Science of Learning from Data, 4th Edition by Alan Agresti, Christine A. Franklin, Bernhard Klingenberg. (Thanks to Aisha Schmidt, student who took the course in 2019 and this year one of the TAs)

### 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 on Tuesday in week x+1, homework due on Monday in the following week. Perhaps for the last homework a different rule applies, this will be announced on time. 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 EDITION 3rd EDITION pages 202-203 pages 216-218 section 8.6 section 8.7 sections 9.1-9.3 sections 9.1-9.2 section 9.4 section 9.3 section 9.5 section 9.4

 FROM WEEK TO WEEK () 1 Online only! Rice, Chapters 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. There may be some additional exercises by the TA (also in subsequent weeks)! Homework: none. 2 Rice, Sections 4.1-4.3 (except Markov inequality, but you can read it yourself). Exercises: Chapter 4: 2, 4, 6, 12, 31, 34*, 46*, 53*, 71(a). Homework: Chapter 4: 32, 45 (for due data consult the TAs) and read the two slides on dependence and correlation. Examples of exam questions: 1, 5, 11, 13, 19(a-d). 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 (for due data consult the TAs). Examples of exam questions: 37(a,e), 40(a,h,i). 4 Rice, sections 8.3-8.5.2 Exercises: chapter 8: 5*(ab), 8, 17*(ab) 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 (for due data consult the TAs). Examples of exam questions: 10, 15, 16 (ignore parts on theory that has not been treated yet). 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. Exercises: chapter 8: 39abc*, 42, 44abc, 49. Homework: chapter 8: 29 (this exercise starts with "Suppose $X_1, X_2,\ldots, X_n$ are i.i.d. $N(\mu, \sigma^2)$ where $\mu$ and $\sigma$ are unknown."), 52; for due data consult the TAs. Examples of exam questions: 10d, 15d,e, 16e,f. 6 Rice, sections 9.2, 9.3, 9.5 (very briefly), perhaps also 9.4. Exercises: chapter 9: 1, 3*, 5, 7*, 9; read the section on the generalized likelihood ratio test. Homework: chapter 9: 11, 16 (for due data consult the TAs). Examples of exam questions: 20, 27. 7 Confidence intervals and testing, regression; Rice, sections 9.4, 14.1-14.4 (emphasis on 14.3, 14.4). Exercises: chapter 9: 15*, chapter 14: 3*, 4, 8*, 9, 12*, 15. Homework: chapter 14: 5, 16, 25* (for due data consult the TAs). Examples of exam questions: 8, 18, 24, 39.

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