6 EC
Semester 1, period 1, 2
5374SIMS6Y
| Owner | Master Mathematics |
| Coordinator | dr. A.J. van Es |
| Part of | Master Stochastics and Financial Mathematics, Master Mathematics, specialization Stochastics , year 1 |
Nowadays simulation methods based on random number generation on powerful computers play an important role in statistics. We highlight two methodologies, the bootstrap and Monte Carlo Markov chain simulation. The bootstrap method has been introduced in 1977 by Bradley Efron. This is a useful, generally applicable, but computationally intensive, method to construct, for instance, confidence intervals. The basic idea of the method is resampling from the original data. The naive bootstrap, paramatric bootstrap and smooth bootstrap shall be discussed.
By running a computer simulated Markov chain for a suitably long time we can generate observations from a distribution close to the stationary distribution of the Markov chain. By choosing suitable transition probabilities practically any distribution can be simulated in this way. We will discuss the Gibbs and Metropolis Hastings algorithms, the basic algorithms for this kind of simulation, as well some of their refinements, bearing in mind the relevance for statistics.
Lecture notes: The Bootstrap, Bert van Es and Hein Putter
Monte Carlo Markov Chain Simulation, Bert van Es
The theory is presented during the lectures. The exercises serve as a way to get a grip on the theory.
|
Activity |
Number of hours |
|
Lectures |
42 |
|
Self study |
66 |
|
Exercises |
66 |
The programme does not have requirements concerning attendance (OER-B).
| Item and weight | Details |
|
Final grade | |
|
75% oral exam | |
|
25% exercises |
The course consists of two parts. The first part, The Bootstrap, has most of the exercises. The second part, MCMC, has less exercises.
The oral exam concerns the second part, MCMC, only.
The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl
| Weeknummer | Onderwerpen | Studiestof |
| 1 | Introduction. Empirical distribution function. | Bootstrap: Chapters 1 and 2 |
| 2 |
Cancelled |
Free |
| 3 |
Different bootstrap methods. Bias and variance bootstrap estimators. |
Bootstrap: Chapter 3 |
| 4 |
Bootstrapping by computer simulation. The Jackknife method.. |
Bootstrap; Chapter 4 (skip Section 4.2), Chapter 5 |
| 5 | Proof that the bootstrap works for the sample mean | Bootstrap: Skip Chapter 6, Do Chapter 7 |
| 6 | Example where the bootstrap fails and the m out on n bootstrap as a remedy. | Bootstrap: Chapter 8 |
| 7 | Accuracy of the bootstrap. Studentized bootstrap, Kernel smoothing. | Bootstrap: Chapter 9 and Appendix C |
| 8 | Free | |
| 9 | The smooth bootstrap: quantile | Bootstrap: Chapter 10 |
| 10 | Introduction and standard simulation methods | MCMC: Chapters 1, 2 |
| 11 | Brief review od standard Markov Chains, Chapter 3, and MCM chains, Chapter 4. Gibbs and Metropolis Hastings samplers. | MCMC: skip Chapter 3. Do Chapter 4 |
| 12 | Chapter 5. One dimensional random number generation by Metropolis Hastings chains. Independent and random walk MH. | MCMC: Chapter 5 |
| 13 | Chapter 6. MCMC for a Bayesian analysis of a contingency table. | MCMC: Chapter 6 |
| 14 | Chapter 7. An example in Finance. Estimation of the parameters of a geometric Brownian motion from asset and option prices. | MCMC: Chapter 7 |
| 15 | ||
| 16 |
The schedule for this course is published on DataNose.
Recommended prior knowledge: Measure Theoretic Probability, Basic statistics.