Modelling of Extreme Events

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

The course is devoted to extremal events models, it means thus events which occur with very low probability, but with significant influence on behaviour of described model. We deal with fluctuation of random sums and fluctuation of maxima, further distributions for modeling extremal events and various models will be introduced and applied. Theoretical results will be applied on real data.

Requirements:

01MIP nebo 01PRST. 01MAS.

Syllabus of lectures:

1. Motivation example from agregated net traffic, solvation (machine learning), on-off approximation.

2. Distribution free inequalities (Cantelli, Chernoff, Hoeffding,...).

3. Nonparametric density estimators and their tails (adaptive kernel estimator, data transformed based methods), semiparametric estimation (Barron).

4. Distribution for modelling extremal values, heavy tailed probability distribution, generalized Pareto, log-gama, log-normal, heavy-tailed Weibull, generalized Gumbel distribution of extremal values - parameter estimates of these distributions and their asymptotic properties.

5. PP and QQ plots for filtration of true distribution of extremal values, ME - mean excess function, its empirical estimator and usage.

6. Return period of the (insuarance) event, ordered statistics, Gumbel method.

7. Fluctuation of random sums, stable and alpha-stable distributions, spectral representation of stable distribution.

8..Fluctuation of random maxima: Gumbel, Fréchet and Weibull distribution as the limit distributions of maximal value of iid variables, large deviations, Fisher-Tippett law.

9. Maximum domain attraction (MDA) - range of stable weak convergence of maxima Mn, application on distribution and expected value of POT models.

10. Models with subexponential distribution for heavy tailed distributions, class of functions R_alpha, with regularly varying of the order alpha in infinity, Karamat theorem.

11. Application on flood data, assurance (cumulative number of insured accidents), many examples.

Syllabus of tutorials:

Study Objective:

Knowledge:

Mathematical models for the events which occur with very low probability, but with significant influence on behaviour of described model - different distributions for modeling extremal events, GEV, GPD, properties of mentioned models, fluctuation of random sums, maximum domain of attraction, POT methods.

Skills:

Application of given methods and models on real data with the aim to predict.

Study materials:

Key references:

[1] P. Embrechts, C. Klüppelberg, T. Mikosch, Modelling Extremal Events, New York Springer, 1997.

Recommended references:

[2] S. Coles, An Introduction to Statistical Modeling of Extreme Values Springer-Verlag London, 2001.

[3] P. Embrechts, H. Schmidli, Modelling of extremal events in insurance and finance, New York, Springer, 1994.

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: