Statistical Methods

The details

Requisites for this module

DIP GN1309 Mathematics and Finance,

MSC GN1312 Mathematics and Finance,

MSC GN1324 Mathematics and Finance,

DIP G20109 Statistics and Operational Research,

MSC G20312 Statistics and Operational Research,

DIP G30009 Statistics,

MSC G30012 Statistics,

DIP N32309 Actuarial Science,

MSC N32312 Actuarial Science

MSC GN1312 Mathematics and Finance,

MSC GN1324 Mathematics and Finance,

DIP G20109 Statistics and Operational Research,

MSC G20312 Statistics and Operational Research,

DIP G30009 Statistics,

MSC G30012 Statistics,

DIP N32309 Actuarial Science,

MSC N32312 Actuarial Science

This Statistical Methods module introduces loss distributions, risk models, copulas, extreme value theory and 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.

Syllabus

1. Decision theory [CS2-1.1,1.2]:

• Loss, risk, admissible and inadmissible decisions, randomised decisions.

• Minimax decisions and Bayes’ solutions, including simple results.

• Explain the concepts of decision theory and apply them.

• Calculate probabilities and moments of loss distributions both with and without limits and risk-sharing arrangements.

• 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.

• Explain the concept of ruin for a risk model.

• Calculate the adjustment coefficient and state Lundberg’s inequality.

• Describe and apply techniques for analysing a delay (or run-off) triangle and projecting the ultimate position.

• Compound distributions and their applications in risk modelling

2. "Monte-Carlo" simulation. [CS1]

3. Bayesian inference [CS1]

• Use Bayes’ theorem to calculate simple conditional probabilities.

• Prior, and posterior distributions. conjugate prior distribution.

• 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.

• Use of Monte Carlo simulation of the posterior distribution to draw inferences.

• Use simple loss functions to derive Bayesian estimates of parameters.

• Bayesian and Empirical Bayes approach to credibility theory and use it to derive credibility premiums in simple cases.

4. Comparative inference [CS1]

• Different criteria for choosing good estimators, tests and confidence intervals.

• Different approaches to inference, including classical, Bayesian and non-parametric.

5. Generalised linear model [CS1]

• Explain the fundamental concepts of a generalised linear model (GLM) and describe how a GLM may apply. In more detail:

• Define an exponential family of distributions. Show that the following distributions may be written in this form: binomial, Poisson, exponential, gamma, normal.

• 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.

• Explain what is meant by the link function and the canonical link function.

• 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.

• Define the deviance and scaled deviance and state how the parameters of a generalised linear model may be estimated.

• Define the Pearson and deviance residuals and describe how they may be used.

• Apply statistical tests to determine the acceptability of a fitted model: Pearson’s chi-square test and the likelihood ratio test

• Fit a generalised linear model to a data set and interpret the output.

6. Copulas [CS2-1.3]:

• Describe how a copula can be characterised as a multivariate distribution function which is a function of the marginal distribution functions of its variates and explain how this allows the marginal distributions to be investigated separately form the dependency between them.

• Explain the meaning of the terms: dependence or concordance, upper and lower tail dependence; and state in general terms how tail dependence can be used to help select a copula suitable for modelling particular types of risk.

• Describe the form and characteristics of the Gaussian copula and the Archimedean family of copulas.

7. Extreme value theory [CS2-1.4]:

• Recognise extreme value distributions, suitable for modelling the distribution of severity of loss and their relationships

• Calculate various measures of tail weight and interpret the results to compare the tail weights.

8. Machine learning [CS2-5.1]:

• Explain the main branches of machine learning and describe examples of the types of problems typically addressed by Machine Learning.

• Explain and apply high-level concepts relevant to learning from data.

• 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.

• Explain in detail and use appropriate software to apply Machine Learning techniques (e.g. penalised regression and decision trees) to simple problems.

• Demonstrate an understanding of the perspectives of statisticians, data scientists, and other quantitative researchers from non-actuarial backgrounds.

9. Liability valuations [CM2-5.1,5.2,5.3]:

Ruin theory

• Explain what is meant by the aggregate claim process and the cash-flow process for a risk.

• Use the Poisson process and the distribution of inter-event times to calculate probabilities of the number of events in a given time interval and waiting times.

• Define a compound Poisson process and calculate probabilities using simulation.

• Define the probability of ruin in infinite/finite and continuous/discrete time and state and explain relationships between the different probabilities of ruin.

• Describe the effect on the probability of ruin, in both finite and infinite time, of changing parameter values by reasoning or simulation.

• Calculate probabilities of ruin by simulation.

Run-off triangles

• Define a development factor and show how a set of assumed development factors can be used to project the future development of a delay triangle.

• Describe and apply a basic chain ladder method for completing the delay triangle using development factors.

• Show how the basic chain ladder method can be adjusted to make explicit allowance for inflation.

• Describe and apply the average cost per claim method for estimating outstanding claim amounts.

• Describe and apply the Bornhuetter-Ferguson method for estimating outstanding claim amounts.

• Describe how a statistical model can be used to underpin a run-off triangles approach.

• Discuss the assumptions underlying the application of the methods of the run-off triangles.

Value basic benefit guarantees using simulation techniques

1. Decision theory [CS2-1.1,1.2]:

• Loss, risk, admissible and inadmissible decisions, randomised decisions.

• Minimax decisions and Bayes’ solutions, including simple results.

• Explain the concepts of decision theory and apply them.

• Calculate probabilities and moments of loss distributions both with and without limits and risk-sharing arrangements.

• 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.

• Explain the concept of ruin for a risk model.

• Calculate the adjustment coefficient and state Lundberg’s inequality.

• Describe and apply techniques for analysing a delay (or run-off) triangle and projecting the ultimate position.

• Compound distributions and their applications in risk modelling

2. "Monte-Carlo" simulation. [CS1]

3. Bayesian inference [CS1]

• Use Bayes’ theorem to calculate simple conditional probabilities.

• Prior, and posterior distributions. conjugate prior distribution.

• 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.

• Use of Monte Carlo simulation of the posterior distribution to draw inferences.

• Use simple loss functions to derive Bayesian estimates of parameters.

• Bayesian and Empirical Bayes approach to credibility theory and use it to derive credibility premiums in simple cases.

4. Comparative inference [CS1]

• Different criteria for choosing good estimators, tests and confidence intervals.

• Different approaches to inference, including classical, Bayesian and non-parametric.

5. Generalised linear model [CS1]

• Explain the fundamental concepts of a generalised linear model (GLM) and describe how a GLM may apply. In more detail:

• Define an exponential family of distributions. Show that the following distributions may be written in this form: binomial, Poisson, exponential, gamma, normal.

• 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.

• Explain what is meant by the link function and the canonical link function.

• 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.

• Define the deviance and scaled deviance and state how the parameters of a generalised linear model may be estimated.

• Define the Pearson and deviance residuals and describe how they may be used.

• Apply statistical tests to determine the acceptability of a fitted model: Pearson’s chi-square test and the likelihood ratio test

• Fit a generalised linear model to a data set and interpret the output.

6. Copulas [CS2-1.3]:

• Describe how a copula can be characterised as a multivariate distribution function which is a function of the marginal distribution functions of its variates and explain how this allows the marginal distributions to be investigated separately form the dependency between them.

• Explain the meaning of the terms: dependence or concordance, upper and lower tail dependence; and state in general terms how tail dependence can be used to help select a copula suitable for modelling particular types of risk.

• Describe the form and characteristics of the Gaussian copula and the Archimedean family of copulas.

7. Extreme value theory [CS2-1.4]:

• Recognise extreme value distributions, suitable for modelling the distribution of severity of loss and their relationships

• Calculate various measures of tail weight and interpret the results to compare the tail weights.

8. Machine learning [CS2-5.1]:

• Explain the main branches of machine learning and describe examples of the types of problems typically addressed by Machine Learning.

• Explain and apply high-level concepts relevant to learning from data.

• 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.

• Explain in detail and use appropriate software to apply Machine Learning techniques (e.g. penalised regression and decision trees) to simple problems.

• Demonstrate an understanding of the perspectives of statisticians, data scientists, and other quantitative researchers from non-actuarial backgrounds.

9. Liability valuations [CM2-5.1,5.2,5.3]:

Ruin theory

• Explain what is meant by the aggregate claim process and the cash-flow process for a risk.

• Use the Poisson process and the distribution of inter-event times to calculate probabilities of the number of events in a given time interval and waiting times.

• Define a compound Poisson process and calculate probabilities using simulation.

• Define the probability of ruin in infinite/finite and continuous/discrete time and state and explain relationships between the different probabilities of ruin.

• Describe the effect on the probability of ruin, in both finite and infinite time, of changing parameter values by reasoning or simulation.

• Calculate probabilities of ruin by simulation.

Run-off triangles

• Define a development factor and show how a set of assumed development factors can be used to project the future development of a delay triangle.

• Describe and apply a basic chain ladder method for completing the delay triangle using development factors.

• Show how the basic chain ladder method can be adjusted to make explicit allowance for inflation.

• Describe and apply the average cost per claim method for estimating outstanding claim amounts.

• Describe and apply the Bornhuetter-Ferguson method for estimating outstanding claim amounts.

• Describe how a statistical model can be used to underpin a run-off triangles approach.

• Discuss the assumptions underlying the application of the methods of the run-off triangles.

Value basic benefit guarantees using simulation techniques

On completion of this module, students should be able to:

• understand loss distributions, with and without risk sharing (1.1)

• understand compound distributions and their applications in risk modelling (1.2)

• basic introductions to copulas (1.3)

• basic introductions to extreme value theory (1.4)

• explain and apply elementary principles of Machine Learning (5.1)

• understand loss distributions, with and without risk sharing (1.1)

• understand compound distributions and their applications in risk modelling (1.2)

• basic introductions to copulas (1.3)

• basic introductions to extreme value theory (1.4)

• explain and apply elementary principles of Machine Learning (5.1)

No additional information available.

The module consist of 31 lectures, 4 classes. In the summer term 3 revision lectures are given.

This module does not appear to have any essential texts. To see non-essential items, please refer to the module's reading list.

Coursework / exam | Description | Deadline | Weighting |
---|---|---|---|

Coursework | Homework 1 | 08/11/2019 | |

Coursework | Homework 2 | 13/12/2019 | |

Exam | 180 minutes during Summer (Main Period) (Main) |

Coursework | Exam |
---|---|

20% | 80% |

Coursework | Exam |
---|---|

0% | 100% |

Module supervisor and teaching staff

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