Modelling Extremal Events

Garant předmětu:

Václav Kůs

Lecturer:

Václav Kůs

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

1.Aggregated traffic in computer nets, possible admission control, machine learning, on-off approximation.

2.Distribution-free inequalities for tail probability estimation, PC simulation of traffic.

3.Nonparametric density estimators and their tails, asymptotic properties, MISE optimality.

4.Semiparametric estimation, retransformed densities, statistical properties, score functions.

5.Phi-divergences, properties, Kolmogorov entropy, Vapnik-Chervonenkis dimension, application.

6.Fluctuation of random sums, stable and α-stable distributions, their characteristics.

7.Generalized central limit theorem, domains of attraction, sub-exponential distributions.

8.Heavy-tail distribution detections, PP and QQ plots, Mean Excess function, its empirical estimator, usage.

9.Return period of (insurance) events, record counting process, Gumbel method of exceedance.

10.Fluctuation of random maxima, Fisher-Tippett law, max-stability, maximum domain of attraction.

11.Generalized extreme value distribution, generalized Pareto distribution, properties and utilization.

12.Estimates of exceedance over threshold, POT methods, estimator of quantile, application.

13.Applications to real data from hydrology, geology, insurance, finance, numerous other examples.

Requirements:

Syllabus of lectures:

Syllabus of tutorials:

Study Objective:

Study materials:

Key references:

[1] R.-D. Reiss, M. Thomas: Statistical Analysis of Extreme Values: from Insurance, Finance, Hydrology and Other Fields, Birkhäuser Basel, 2014.

[2] S. Foss, D. Korshunov, S. Zachary: An Introduction to Heavy-Tailed and Subexponential Distributions, Springer-Verlag, New York, 2013.

Recommended references:

[3] N. Markovich: Nonparametric Analysis of Univariate Heavy-Tailed Data, Wiley, 2007.

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

[5] S. Coles: An Introduction to Statistical Modeling of Extreme Values, Springer, London, 2001.

Note:

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikované matematicko-stochastické metody (compulsory elective course)