Simulation Methods in Statistics

Course manual 2024/2025

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 as  some of their refinements, bearing in mind the relevance for statistics. Other simulation based statistical methods will also be briefly discussed.

The emphasis of this course is to provide some basic understanding and knowledge of simulation methods used in statistics and illustrate them through simple examples. Secondly, the course will shed light on the theory behind these techniques, which helps to understand why they work and when they would not work. We will not get into the details of programming itself and will not deal with specialized R packages. All the examples and exercises can be handled with simple R functions.

Study materials

Syllabus

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

• Lecture notes: Monte Carlo Markov Chain Simulation, Bert van Es

Software

• R

Other

• Any additional material on topics about other simulation based statistical methods  will be provided as pdf or accessible via UvA library.

Objectives

• be able to explain the bootstrap and the MCMC method, in general and in certain applications;
• be able to apply the bootstrap method in theoretical exercises;
• be able to apply the bootstrap and the MCMC method 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 assess which simulation method (among the ones that will be studied) is appropriate for solving a given problem;
• be able to assess whether a particular simulation method would work for a (simple) given problem based on the theory behind it.
• be able to explain the main ideas behind other simulation methods used in statistics such as the multiple imputation, permutation approach, SIMEX, cross-validation;

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

Attendance

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

Assessment

Apart from the two graded assignments described below, there is a final written exam (only theoretical questions or in the form of pseudocode) which counts for 70% of the grade. Exam grade needs to be at least 5 in order to pass the course. 

The assignments do not count towards the resit exam, i.e. the resit counts for 100% of the final grade and it will include some practical (R) exercise as a replacement for Assignments 1-2.

For the exam and resit it is allowed to have one A4-sized double-sided handwritten cheat sheet.

Inspection of assessed work

Contact the course coordinator to make an appointment for inspection.

Assignments

Assignment 1

• The bootstrap method 

Assignment 2

• MCMC Simulation

The two assignments are individual and count each for 15% of the final grade. They consist of exercises that include mainly practical (R) questions. 

Assignments are graded on a scale 1-10, without any minimum grade requirement. Assignments that are not handed in count as  having grade 1. 

Other theoretical exercises are suggested as practice exercises (not graded) and will be partially discussed in class. 

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

Additional information

Measure Theoretic Probability, basic statistics, basic familiarity with R (an R manual can be found for example here: https://cran.r-project.org/manuals.html).

Contact information

Coordinator

• dr. E. Musta