(Created 2011-09-01.)

STATISTICAL MODELLING OF MULTIVARIATE EXTREME VALUES | FMSN15 |

**Aim**

Multivariate extreme values occure in, e.g., economy, safety and reliability, insurance mathematics, hydrology, meteorology. environmental sciences, och ocenanography. They often show complicated dependencies between several variables, e.g. between wind speed, wind direction, wave height and ocean currents. This calls for special methods that can be used, e.g., for analysis of trends, calculation of flooding risks, and modelleling storm damage, corrosion speed, or financial risks. Climat and environmental changes, as well as an increasingly complicated financial market, pose new demands on deapend knowledge in these fields. This course is a countinuation of FMS155 Statistical Modelling of Extreme Values, and teaches methods for analysis of multivariate and spatial extreme values.

*Knowledge and understanding*

For a passing grade the student must

- describe how to define extreme values for multivariate samples,
- describe different characterisations of multivariate extreme value distributions and the relationship between them,
- explain how to generalize the "peaks over threshold"-model to higher dimensions and which asymptotic distributions arise,
- explain which statistical methods can be used for the analysis of extreme values.

*Skills and abilities*

For a passing grade the student must

- handle multivariate data for analysis of extreme values,
- fit extreme value distribution using different methods,
- validate the valitidy of the extreme value model and make suitable modifications of the model,
- use the resulting model for prediction,
- use a statistical computer program for analysis of data,
- present the analysis and conclusions of a practical problem in a written report.

*Judgement and approach*

For a passing grade the student must

- always check the prerequisites befor stating an extreme value model,
- evaluate the plausibility of a performed study,
- reflect over the limitations of the chosen model and estimation method, as well as alternative solutions.

**Contents**

Weak convergence for normalized extreme values of stochastic vectors, different characterisations of multivariate extreme value distributions, "peaks over threshold"-model in the multivariate case, different definitions of multivariate generalized Pareto distributions, statistical inference for multivariate extreme values, parametric and semi-parametric methods for multivariate extreme values, use of copula in modelling extreme values, point process characterisation of extreme values, prediction of extreme values, examples of applications of the theory, e.g., estimation of operational risk, climate changes and wind insurances.

**Literature**

Resnick, S.I. (2007) Extreme values, Regular Variation and Point Processes, Berlin: Springer-Verlag.

Harry, J. (1997) Multivariate Models and Multivariate Dependence Concepts, Chapman & Hall/CRC Monographs on Statistics & Applied Probability.

**Code: **0111.
**Name: **Written Examination.

**Higher education credits: ** 6.
**Grading scale: **TH.
**Assessment:** Written examination.

**Code: **0211.
**Name: **Laboratory Work and Home Assignments.

**Higher education credits: ** 1,5.
**Grading scale: **UG.
**Assessment:** Laboratory work and home assignments.