Simulation Methods in Statistics

6 EC

Semester 1, period 1, 2

5374SIMS6Y

Owner Master Mathematics
Coordinator prof. dr. J.M. Mooij
Part of Master Mathematics, Master Stochastics and Financial Mathematics, Master Mathematics, specialization Stochastics , year 1Master Stochastics and Financial Mathematics, specialization Data Analysis and Statistics, year 1Master Mathematics, specialization Algebra and Geometry, year 1Master Mathematics, specialization Analysis and Dynamical Systems, year 1Master Mathematics, specialization Mathematical Physics, year 1Master Stochastics and Financial Mathematics, specialization Financial Mathematics, year 1Master Stochastics and Financial Mathematics, specialization (Applied) Probability and Decision Making, year 1

Course manual 2020/2021

Course content

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, parametric 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.

Study materials

Syllabus

  • Lecture notes: The Bootstrap, Bert van Es and Hein Putter

  • Monte Carlo Markov Chain Simulation, Bert van Es

Objectives

  • be able to explain the bootstrap method;
  • be able to apply the bootstrap method in simple theoretical exercises;
  • be able to apply the bootstrap on the computer in practical situations;
  • be able to explain and apply several standard computer simulation methods like inversion and acceptance/rejection sampling;
  • be able to explain the MCMC method, in general and in certain applications;
  • be able to apply the MCMC method on the computer in practical situations.

Teaching methods

  • Lecture
  • Exercises

The theory is presented during the lectures. The exercises serve as a way to get a grip on the theory (most will not be graded).

Learning activities

Activity

Number of hours

Lectures

28

Self study

66

Exercises

66

 

Attendance

This programme does not have requirements concerning attendance (TER-B).

Assessment

Item and weight Details

Final grade

85%

Tentamen

5%

The Bootstrap: Assignment 1

5%

MCMC Simulation: Assignment 2

5%

MCMC Simulation: Assignment 3

Fraud and plagiarism

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

Course structure

Week number
 Topics Chapters Exercises
1 Introduction. Empirical distribution function.  Bootstrap: Chapters 1 and 2 1
2

Different bootstrap methods. Bias and variance bootstrap estimators.

  Bootstrap: Chapter 3 2
3

Bootstrapping by computer simulation.

The Jackknife method.

  Bootstrap; Chapter 4, Chapter 5 3
4 Proof that the bootstrap works for the sample mean  Bootstrap: Chapter 6, Chapter 7 7, Assignment 1
5 Example where the bootstrap fails and the m out on n bootstrap as a remedy.  Bootstrap: Chapter 8 6
6 Accuracy of the bootstrap. Studentized bootstrap, Kernel smoothing.  Bootstrap: Chapter 9 and Appendix C 10
7 The smooth bootstrap  Bootstrap: Chapter 10 and Appendix B 8,9
8      
9

Introduction and standard simulation methods.

  MCMC: Chapters 1, 2  
10

Chapter 3. Brief review of standard Markov Chains.

Chapter 4. MCMC chains, Gibbs and Metropolis Hastings samplers.

  MCMC: Chapters 3, 4 Assignment 2
11 Chapter 5.  MCMC examples in univariate random number generation.  MCMC: Chapter 5 Assignment 3
12 Chapter 6. Bayesian analysis of a contingency table.  MCMC: Chapter 6  
13 Chapter 7. An example in finance.  MCMC: Chapter 7  
14      
15      

 

Timetable

The schedule for this course is published on DataNose.

Additional information

Recommended prior knowledge: Basic statistics.

Contact information

Coordinator

  • prof. dr. J.M. Mooij