MA318-7-AU-CO:
Statistical Methods
2020/21
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Autumn
Postgraduate: Level 7
Current
Thursday 08 October 2020
Friday 18 December 2020
15
16 July 2020
Requisites for this module
(none)
(none)
(none)
(none)
MA322
DIP GN1309 Mathematics and Finance,
MSC GN1312 Mathematics and Finance,
MSC GN1324 Mathematics and Finance,
DIP G20109 Optimisation and Data Analytics,
MSC G20312 Optimisation and Data Analytics,
DIP G30009 Statistics,
MSC G30012 Statistics,
DIP N32309 Actuarial Science,
MSC N32312 Actuarial Science,
MPHDG20048 Operational Research,
PHD G20048 Operational Research,
MPHDG30048 Statistics,
PHD G30048 Statistics,
MPHDN32348 Actuarial Science,
PHD N32348 Actuarial Science
The module introduces decision theory, loss distributions, risk modelling, "Monte Carlo" simulation, Bayesian inference, comparative inference, the generalised linear model and basic contents of machine learning.
This module covers 15% of required material for the Institute and Faculty of Actuaries CS1 syllabus and 30% (CS2 Units 1-4 & 18) of required material for the Institute and Faculty of Actuaries CS2.
The aims of this module are:
1. to learn the concept and basic principle of Bayesian inference.
2. to learn the concept of decision theory.
3. to learn the principles and methods of choosing good estimators.
4. to learn the basic concept of ruin theory.
5. to learn concepts and how to implement “Monte-Carlo” simulation with R.
6. to learn concepts and how to implement generalised linear model with R.
7. to learn concepts of machine learning and explore it in R.
On completion of this module, students should be able to:
• Understand concepts of decision theory;
• Apply concepts of decision theory (risk models);
• Understand techniques for analysing a delay (or run-off) triangle and projecting the ultimate position;
• Understand "Monte-Carlo" simulation;
• Understand basic principles of Bayesian inference;
• Understand principles and methods to choose good estimators;
• Understand basic concepts of a generalised linear model;
• Understand elementary principles of Machine Learning;
• Apply R to do GLM, Bayesian simulation and machine learning.
Syllabus
Bayesian Statistics (including decision theory and extreme value theory) [CS1, CS2-1.1,1.2, CS2-1.4]
1. Use Bayes' theorem to calculate simple conditional probabilities.
2. Prior, posterior distributions, and conjugate prior distribution.
3. Choice of prior: bets, conjugate families of distributions, vague and improper priors. Predictive distributions. Bayesian estimates and intervals for parameters and predictions. Bayes factors and implications for hypothesis tests.
4. Use of Monte Carlo simulation of the posterior distribution to draw inferences.
5. Loss functions and Bayesian estimates, including explain the concepts of decision theory and apply them; loss, risk, admissible and inadmissible decisions, randomised decisions; minimax decisions and Bayes' solutions, including simple results; calculate probabilities and moments of loss distributions.
6. Bayesian and Empirical Bayes approach to credibility theory and use it to derive credibility premiums in simple cases.
7. Understand extreme value distributions, suitable for modelling the distribution of severity of loss and their relationships; understand how to use extreme value.
Ruin theory [CM2-5.1,5.2,5.3]
1. Construct risk models involving frequency and severity distributions and calculate the moment generating function and the moments for the risk models both with and without simple reinsurance arrangements.
2. Explain the concept of ruin for a risk model.
3. Calculate the adjustment coefficient and state Lundberg's inequality.
4. Compound distributions and their applications in risk modelling
5. Explain what is meant by the aggregate claim process and the cash-flow process for a risk.
6. Define a compound Poisson process and calculate probabilities using simulation.
7. Define the probability of ruin in infinite/finite and continuous/discrete time and state and explain relationships between the different probabilities of ruin.
8. Describe the effect on the probability of ruin, in both finite and infinite time, of changing parameter values by reasoning or simulation.
Predictive modelling [CS1, CS2-5.1]
1. Generalised linear model: fundamental concepts of (GLM)
2. Define an exponential family of distributions. Show that the following distributions may be written in this form: binomial, Poisson, exponential, gamma, normal.
3. State the mean and variance for an exponential family and define the variance function and the scale parameter. Derive these quantities for the distributions above.
4. Explain what is meant by the link function and the canonical link function.
5. Explain what is meant by a variable, a factor taking categorical values and an interaction term. Define the linear predictor, illustrating its form for simple models, including polynomial models and models involving factors.
6. Define the deviance and scaled deviance and state how the parameters of a generalised linear model may be estimated.
7. Define the Pearson and deviance residuals and describe how they may be used.
8. Apply statistical tests to determine the acceptability of a fitted model: Pearson's chi-square test and the likelihood ratio test
9. Fit a generalised linear model to a data set and interpret the output.
10. Describe and apply techniques for analysing a delay (or run-off) triangle and projecting the ultimate position, under GLM (how GLM underpin run-off triangle methodology). Describe and apply a basic chain ladder method for completing the delay run-off triangle using development factors, for estimating outstanding claim amounts.
11. Machine learning: explain the main branches of machine learning and describe examples of the types of problems typically addressed by Machine Learning.
12. Describe and give examples of key supervised and unsupervised Machine Learning techniques, explaining the difference between regression and classification and between generative and discriminative models.
13. Explain in detail and use appropriate software to apply Machine Learning techniques (e.g. penalised regression and decision trees) to simple problems.
14. Demonstrate an understanding of the perspectives of statisticians, data scientists, and other quantitative researchers from non-actuarial backgrounds.
Teaching will be delivered in a way that blends face-to-face classes, for those students that can be present on campus, with a range of online lectures, teaching, learning and collaborative support.
This module does not appear to have any essential texts. To see non-essential items, please refer to the module's reading list.
Assessment items, weightings and deadlines
| Coursework / exam |
Description |
Deadline |
Coursework weighting |
| Coursework |
Test |
|
|
| Exam |
Main exam: 240 minutes during Summer (Main Period)
|
Exam format definitions
- Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
- In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
- In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
- In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary,
for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.
Your department will provide further guidance before your exams.
Overall assessment
Reassessment
Module supervisor and teaching staff
Dr Yanchun Bao, email: ybaoa@essex.ac.uk.
Dr Yanchun Bao & Dr Peng Liu
Dr Yanchun Bao (ybaoa@essex.ac.uk), Dr Peng Liu (peng.liu@essex.ac.uk)
Yes
No
No
Dr Dimitrina Dimitrova
Cass Business School, City, University of London
Senior Lecturer
Available via Moodle
Of 2046 hours, 2 (0.1%) hours available to students:
2044 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).
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