ST3456: Modern Statistical Methods II

Module Code: ST3456 (descriptor)

Module Title: Modern Statistical Methods II (Simulation methods)

Cohorts: Maths 

Week range: 03-08, 10-14

Semester: S1 (Michaelmas) 

Total hours: 33 (28 classes + 5 labs)


Timetable: 

- Monday, h. 12-14, room LB01

- Tuesday, h. 10-11, room LB 1.07 (or ICT1 lab when labs are scheduled)


Textbook: 

  • Lecture notes will be uploaded to this webpage

Other references:

  • Ross, "Simulation"
  • Rubinstein & Kroese, "Simulation and Monte Carlo Method"
  • Robert & Casella, "Monte Carlo statistical methods"
  • Robert & Casella, "Introducing Monte Carlo methods with R"
  • Devroye, "Non-uniform random variate generation"


Links:

Lecture notes (password protected):

  • 10/09/18: Section 1: motivating examples (pdf)
  • 11/09/18: Section 2.1: inverse transform method (pdf)
  • 17/09/18: Section 2.2: transformation methods (pdf)
  • 18/09/18: Section 2.3: acceptance-rejection method (pdf)
  • 01/10/18: Section 3, 3.1: controlling the variance; common and antithetic variables (pdf)
  • 08/10/18: Section 3.2, 3.3.1: control variables; conditional expectation (pdf)
  • 15/10/18: Section 3.3.2: variance reduction by conditioning (pdf)
  • 15/10/18: Lecture notes first 6 weeks: Sections 1, 2 and 3 (pdf)


Problem sets (password protected):
  • 24/09/18: problem set 1 (pdf) (optional: solutions can be handed in by the 8/10/18)

Slides (password protected):
  • 10/09/18: Class 1 (pdf)

Labs (password protected):
  • 25/09/18: Lab 1 - instructions (pdf), R script (R) and solutions (R)
  • 02/10/18: Lab 2 - instructions (pdf), R script (R) and solutions (R)
  • 09/10/18: Lab 3 - instructions (pdf), R script (R) and solutions (R)
  • 16/10/18: Lab 4 - instructions (pdf), R script (R) and solutions (R)


Syllabus (lecture by lecture):

  • 10/09/18 (2 h): motivating examples: Monte Carlo integration, Bayesian Inference, analysis of leukaemia data.
  • 11/09/18 (1 h): inverse transform method (with proof), ITM for exponential random variables (example 2.5), max and min of IID random variables (first method).
  • 17/09/18 (2 h): max and min of IID random variables (ITM method), ITM for discrete random variable, Transformation methods, Box-Muller method.
Assignment: see example 2.7 (ITM for Gamma(n, lambda)).
  • 18/09/18 (1 h): acceptance-rejection (AR) method, with proof.
  • 24/09/18 (2 h): geometric distribution for the number of needed iterations in AR, AR for densities known up to a constant, example: optimal choice of exponential envelope for gamma target density.
  • 25/09/18 (1 h): lab 1 - ITM for discrete random variables (Binomial and Poisson).
  • 01/10/18 (2 h): Monte Carlo estimation of expected value, mean squared error (MSE) and relative error, principle of variance reduction, estimating probability of rare event via crude Monte Carlo (example 3.1), antithetic variables (example 3.2), common variables (example 3.3).
  • 02/10/18 (1 h): lab 2 - ITM (Exponential) and AR (Gamma).
  • 08/10/18 (2 h): Control variables, conditional expectation, conditional variance formula.
  • 09/10/18 (1 h): lab 3 - Simlating Normals via AR and Box-Muller.
  • 15/10/18 (2 h): Variance reduction by conditioning, Example: comparing methods in estimating pi.
  • 16/10/18 (1 h): lab 4 - Integral estimation via simulation


Videos:
  • 18/09/18: acceptance-rejection for bounded density on compact support:

AR_movie.mov


  • 18/09/18: acceptance-rejection for truncated Normal(0,1) with Exponential(1) envelope:

AR_movie_HALF.mov