# MMath Mathematics with Financial Mathematics

Year of entry: 2023

## Course unit details:Statistical Computing

Unit code MATH48091 15 Level 4 Semester 1 Department of Mathematics No

### Overview

Computers are an invaluable tool to modern statisticians. The increasing power of computers has greatly increased the scope of inferential methods and the type of models which can be analysed. This has led to the development of a number of computationally intensive statistical methods, many of which will be introduced in this course.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Probability 2 MATH20701 Pre-Requisite Compulsory
Statistical Methods MATH20802 Pre-Requisite Compulsory
Practical Statistics MATH20811 Pre-Requisite Recommended
Regression Analysis MATH38141 Pre-Requisite Recommended
MATH20701 and MATH20802 are pre-requisites for MATH48091

MATH20812 Practical Statistics   (Recommended, but not compulsory)

MATH38141 Regression Analysis   (Recommended, but not compulsory)

Students are not permitted to take MATH48091 and MATH68091 for credit in an undergraduate programme and then a postgraduate programme.

### Aims

To introduce the student to computational statistics, both the underlying theory and the practical application.

### Learning outcomes

On successful completion of this course unit students will be able to:

• construct algorithms to simulate random observations from probability distributions using a variety of methods and explain mathematically why they work;
• construct and derive the statistical properties of Monte Carlo estimators, as well as alternatives which seek to reduce variance;
• apply the bootstrap to assign measures of accuracy to sample estimates and to derive their statistical properties analytically in some simple cases;
• recognise a non-linear regression model and be able to formulate the Gauss-Newton algorithm to find the parameter estimates from data;
• use the EM algorithm to find maximum likelihood estimators of parameters in some given contexts when we have incomplete sample information;
• implement the methodology discussed in the module (and also carry out simple simulation studies) on a computer using the statistical software R.  To present informatively and discursively the results of computations.

### Syllabus

• Introduction [1]
• Simulating random variables: inversion of the cdf; rejection sampling; transformations; ratio of uniforms. [4]
• Monte Carlo integration [1]
• Variance Reduction: importance sampling; control variates. [2]
• Nonparametric bootstrap methods; the Jackknife. [6]
• Nonlinear regression: model specification; least squares estimation; Gauss-Newton algorithm. [2]
• EM algorithm: data augmentation; the multinomial model; mixture distributions, censored data, Monte-Carlo EM. [6]

### Assessment methods

Method Weight
Other 50%
Written exam 50%
• Three pieces of coursework each worth 16.67%: 50%
• End of semester written examination: 50%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework also provides an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• Rizzo, M.  Statistical Computing with R.  Chapman & Hall
• Ripley, B.D.  Stochastic Simulation.  Wiley.
• Efron, B. and Tibshirani, R. An introduction to the bootstrap.  Chapman & Hall

### Study hours

Scheduled activity hours
Lectures 24
Practical classes & workshops 22
Tutorials 12
Independent study hours
Independent study 92

### Teaching staff

Staff member Role
Georgi Boshnakov Unit coordinator