### Module Details

 The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
 Title STATISTICAL METHODS IN ACTUARIAL SCIENCE Code MATL374 Coordinator Dr S Mitra Mathematical Sciences Sovan.Mitra@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2019-20 Level 6 FHEQ First Semester 15

### Aims

1.Provide a solid grounding in analysis of general insurance data, Bayesian credibility theory and the loss distribution concept.

2. Provide an introduction to statistical methods for managing risk in non-life insurance and finance.

3. Prepare the students adequately to sit for the exams of CT6 subject of the Institute of actuaries.

### Learning Outcomes

(LO1) Be able to apply the estimationmethods described in (b) of the Syllabus for the distribution described in (a)of the Syllabus, be able to make hypothesis testing described in (b) of theSyllabus for the distribution described in (a) of the Syllabus.

(LO2) Be able to estimate the parameters ofthe loss distributions when data complete/incomplete using the method of moments and the method of maximum likelihood, be able tocalculate the loss elimination ratio.

(LO3) Understand and use the Buhlmann model, the Buhlmann-Straub model, beable to state the assumptions of the GLM models – normal linear model,understand the properties of the exponential family.

(LO4) Be able to express the values of thelife assurances in (d) of the Syllabus and the life annuities in (f) of theSyllabus in terms of the life table functions. Be able to use approximationsfor the evaluation of the life assurances in (d) of the Syllabus and the lifeannuities in (f) of the Syllabus based on a life table.

(LO5) Be able to describe the properties ofa time series using basic analytical and graphical tools.

(LO6) Understand the definitions,properties and applications of well know time seriesmodels, fit time series models to practical data sets and select the suitablemodels, be able to perform simple statistical inference (forecasting) based onthe fitted models, estimate and remove possible trend and seasonality in a timeseries, analyse the residuals of a time series using stationary models.

(S1) Problem solving skills

(S2) Numeracy

### Syllabus

(a) Review of probability theory and Statistical inference Estimation, method of moments, unbiasedness, the likelihood function and maximum likelihood estimation, confidence interval for the population mean difference, hypothesistesting, the exponential, gamma, log-gamma, Pareto, generalized Pareto, normal, lognormal,Weibull, Burr, log-logistic, Benktander IIdistributions, comparison of the tails of distributions (asymptotictechniques), mixing distributions.

(b) Loss distribution Moment generating functions (if exists), moments, the variance and alternative methods for the calculation of moments when the moment generating function doesnot converge for the distributions described in (a) of the Syllabus, the Kolmogorov – Smirnov test, the likelihood ratio test.

(c) Special types of insurance schemes &Reinsurance Define simple insurance schemes due to deductibles and retention of limits,policy limits, proportional reinsurance, excess of loss reinsurance,statistic al inference for the previous insurance/reinsurance schemes with themethods in the in (b) of the Syllabus. (d) Bayesian statistics and credibility theory The Bayes Theorem and conditional probabilities, the prior and the posterior distribution, conjugate prior distributions and the linear exponential family,the loss function, the credibility premium, the credibility factor, the Buhlmann model, the Buhlmann-Straub model, exact credibility.

(e) GLM Linear regression, the multiple linear regression, the normal linear model,GLM: the exponential family of distributions for the binomial, Poisson,exponential, Gamma, normal distribution, the Pearson and deviance residuals,application of tests in model fitting.

(f) Simulation The basic of simulation, the simulation approach, the truly random and thepseudo random numbers, derivation/generation of the pseudo random numbers,applications to sets of simulation, Monte Carlo simulation, the number ofsimulations .

(g) Time Series with applications Serial dependence (autocorrelation coefficients, correlograms), stationary and non-stationary processes, filters and operators, Moving Average (MA) process(properties and applications), Autoregressive (AR) process (properties andapplications), Autoregressive moving average (ARMA) process (properties and applications), extension to integrated (ARIMA) processes, non-linear processes,such as ARCH and GARCH, the multivariate autoregressive model,

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Written Exam This is an anonymous assessment. Assessment Schedule (When) :End of Semester  150 minutes    100
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes