|
|
Coupling and speed of convergence of discrete MCMC algorithms.
Kalaš, Martin ; Prokešová, Michaela (advisor) ; Dvořák, Jiří (referee)
Convergence of the marginal distribution of a Markov chain to its stationary distribution is an essential property of this model with many applications in different fields of modern mathematics. Such typical applications are for example the Markov Chain Monte Carlo algorithms, which are useful for sampling from complicated probability distributions. A crucial point for usefulness of such algorithms is the so called mixing time of corresponding Markov chain, i.e. the number of steps the chain has to make for the difference between its current marginal distribution and stationary distribution to be sufficiently small. The main goal of this thesis is to describe a method for estimation of the mixing time based on a probability technique called coupling. In the first part we collect some definitions and propositions to show how the method works. Later the method is demonstrated on several traditional examples of Markov chains including e.g. random walk on a graph. In the end we study Metropolis chain on the set of proper colorings of a graph as a specific example of MCMC algorithm and show how to estimate its mixing time.
|
|
|
Modelling Bonus - Malus Systems
Stroukalová, Marika ; Mazurová, Lucie (advisor) ; Prokešová, Michaela (referee)
Title: Modelling Bonus - Malus Systems Author: Marika Stroukalová Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Lucie Mazurová, Ph.D., KPMS MFF UK Abstract: In this thesis we deal with bonus-malus tariff systems commonly used to adjust the a priori set premiums according to the individual claims during mo- tor third party liability insurance. The main aim of this thesis is to describe the standard model based on the Markov chain. For each bonus-malus class we also determine the relative premium ("relativity"). Another objective of this thesis is to find optimal values for the relativities taking into account the a priori set premiums. We apply the theoretical model based on the stationary distribu- tion of bonus-malus classes on real-world data and a particular real bonus-malus system used in the Czech Republic. The empirical part of this thesis compares the optimal and the real relativities and assesses the suitability of the chosen theoretical model for the particular bonus-malus system. Keywords: bonus-malus system, a priori segmentation, stationary distribution, relativity, quadratic loss function 1
|
|
|
Multi - event Bonus - Malus System
Kaplanová, Martina ; Mazurová, Lucie (advisor) ; Branda, Martin (referee)
This work deals with bonus - malus systems for automobile insurance that distinguishtypes of claim. The first part of this work is definition of bonus - malus systems that do not distinguish types of claim and then their expansion just to multi - event bonus - malus systems. The main focus of the work is computation of stationary distribution for different systems, which means the distribution of classes in which the system stabilizes. Furthermore, there are several simulations of trajectory of insured through the system based on the number and type of accidents that they have caused. Finally, relative frequencies of classes in which insured is at the end of the simulation and the stationary distribution of the system are compared. Powered by TCPDF (www.tcpdf.org)
|
|
|
Applications of Markov chains
Berdák, Vladimír ; Beneš, Viktor (advisor) ; Kadlec, Karel (referee)
The goal of the thesis is the use of Markov chains and applying them to algorithms of the method Monte Carlo. Necessary theory of Markov chains is introduced and we are aiming to understand stationary distribution. Among MCMC methods the thesis is focused on Gibbs sampler which we apply to the hard-core model. We subsequently simulate distribution of ones and zeros on vertices of a graph. Statistical characteristics of the number of ones are estimated from realizations of MCMC and presented in figures.
|
|
|
Modelling Bonus - Malus Systems
Stroukalová, Marika ; Mazurová, Lucie (advisor) ; Prokešová, Michaela (referee)
Title: Modelling Bonus - Malus Systems Author: Marika Stroukalová Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Lucie Mazurová, Ph.D., KPMS MFF UK Abstract: In this thesis we deal with bonus-malus tariff systems commonly used to adjust the a priori set premiums according to the individual claims during mo- tor third party liability insurance. The main aim of this thesis is to describe the standard model based on the Markov chain. For each bonus-malus class we also determine the relative premium ("relativity"). Another objective of this thesis is to find optimal values for the relativities taking into account the a priori set premiums. We apply the theoretical model based on the stationary distribu- tion of bonus-malus classes on real-world data and a particular real bonus-malus system used in the Czech Republic. The empirical part of this thesis compares the optimal and the real relativities and assesses the suitability of the chosen theoretical model for the particular bonus-malus system. Keywords: bonus-malus system, a priori segmentation, stationary distribution, relativity, quadratic loss function 1
|
|
|
Coupling and speed of convergence of discrete MCMC algorithms.
Kalaš, Martin ; Prokešová, Michaela (advisor) ; Dvořák, Jiří (referee)
Convergence of the marginal distribution of a Markov chain to its stationary distribution is an essential property of this model with many applications in different fields of modern mathematics. Such typical applications are for example the Markov Chain Monte Carlo algorithms, which are useful for sampling from complicated probability distributions. A crucial point for usefulness of such algorithms is the so called mixing time of corresponding Markov chain, i.e. the number of steps the chain has to make for the difference between its current marginal distribution and stationary distribution to be sufficiently small. The main goal of this thesis is to describe a method for estimation of the mixing time based on a probability technique called coupling. In the first part we collect some definitions and propositions to show how the method works. Later the method is demonstrated on several traditional examples of Markov chains including e.g. random walk on a graph. In the end we study Metropolis chain on the set of proper colorings of a graph as a specific example of MCMC algorithm and show how to estimate its mixing time.
|
guest ::
