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Dependent zeros
Hanousek, Jan ; Pešta, Michal (advisor) ; Hendrych, Radek (referee)
This thesis investigates a specific type of non-negative time series containing a sig- nificant proportion of zeros. The goal of this work is to create a stochastic model which would be an appropriate representation of such time series. After examining existing theory about stochastic processes and the estimation of their parameters, we propose our own final models. Their suitability is tested using real-world data and the procedure shows that each model has its own advantages and limitations. Overall, the results are satisfactory, proving the credibility of the models and their applicability in practice and paving the way for possible further research on this topic. 1
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Operational risk and marked Poisson process
Váchová, Karla ; Pešta, Michal (advisor) ; Dvořák, Jiří (referee)
The subject of this bachelor thesis entitled "Operational risk and marked Poisson process" is the modelling of operational risk using marked Poisson process. The Poisson process is a type of a point process that models randomly distributed points on some underlying space. Because of its mathematical properties, it is a quite frequently used model in biology, astronomy, ecology or economics, for example. This bachelor thesis describes its basic properties and uses the marked Poisson process to model loss frequency and severity belonging to bank's operational risk. 1
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Truncated random vectors
Raab, Petr ; Pešta, Michal (advisor) ; Komárek, Arnošt (referee)
This bachelor thesis deals with truncated random vectors, distributions and properties of theirs. Truncated random vectors theory is then used to solve problem of delayed reporting of non-life insurance claims. At the of this thesis there are shown properties and behaviour of the estimators, which are constructed in this thesis, while being applicated on real life data from vehicle accident insurancy. 1
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Copulae for non-continuous distributions
Mifkovič, Matej ; Pešta, Michal (advisor) ; Omelka, Marek (referee)
Copulas are a popular choice when assessing the dependence structure between continuous random variables. However, major difficulties arise as soon as one of the random variables is non-continuous. This thesis introduces the basics of copula theory based on the cited literature. The main focus of this thesis is to introduce the reader to the field of non- continuous copula modelling and highlight all major issues. At the same time, empirical evidence with discussion is presented to suggest that copula modelling and inference may be a viable option when additional care and caution are applied. Afterwards, accumulated theoretical knowledge is demonstrated on real-world data concerning bike-sharing.
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Causality in multiple time series
Kusenda, Ondrej ; Pešta, Michal (advisor) ; Cipra, Tomáš (referee)
The bachelor thesis describes causality in multiple time series. Mul- tiple time series are formulated by using vector autoregressive models (VAR). The general properties of the VAR model are defined in the thesis . Model cre- ation involves VAR order selection, estimation of its parameters and checking the properties of the VAR model. The basic concepts of the Granger and instan- taneous Granger causality and theorems for the classification of these relations are defined in the thesis. Tests for the Granger and instantaneous Granger causality are described on suitable models. Subsequently, theoretical knowledge is applied to real data, which are available in the database in the program R. The practical part of the bachelor thesis is performed in the program R. 1
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Difference and differential equations in life insurance
Kirešová, Katarína ; Kříž, Pavel (advisor) ; Pešta, Michal (referee)
The diploma thesis deals with the calculation of life insurance reserves, higher mo- ments and the distribution function of future payments of reserves using difference and differential equations. In the beginning, the basic theory of a stochastic process, insu- rance model, cash flow, and reserve is summarized. After that, equations themselves are derived; first in general and then for four specific types of insurance. Subsequently, a cal- culation of premiums is presented for each type of insurance. The next two chapters deal with the calculation of higher moments and the distribution function. After deriving the formulas for four types of insurance, the reserves, standard deviations, and distribution functions are calculated for specific values and then they are compared with the Monte Carlo simulation. The conclusion contains pros and cons of the method compared to the simulation. 1
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Gradual change model
Míchal, Petr ; Hlávka, Zdeněk (advisor) ; Pešta, Michal (referee)
The thesis aims at change-point estimation in gradual change models. Methods avail- able in literature are reviewed and modified for point-of-stabilisation (PoSt) context, present e.g. in drug continuous manufacturing. We describe in detail the estimation in the linear PoSt model and we extend the methods to quadratic and Emax model. We describe construction of confidence intervals for the change-point, discuss their interpre- tation and show how they can be used in practice. We also address the situation when the assumption of homoscedasticity is not fulfilled. Next, we run simulations to calculate the coverage of confidence intervals for the change-point in discussed models using asymp- totic results and bootstrap with different parameter combinations. We also inspect the simulated distribution of derived estimators with finite sample. In the last chapter, we discuss the situation when the model for the data is incorrectly specified and we calculate the coverage of confidence intervals using simulations. 1
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Technical reserves of non-life insurance in the internal solvency models
Thomayer, Jiří ; Mertl, Jakub (advisor) ; Pešta, Michal (referee)
Title: Technical reserves of non-life insurance in the internal solvency model Author: Bc. Jiří Thomayer Department: Department of Propability and Mathematical Statistics Supervisor: Mgr. Ing. Jakub Mertl Abstract: In this work we study and describe calculation of solvency capital using the standard formula contained in the Directive of the European Union (Solvency II), which should be put into practice in Europe on 1 January 2013. This calcu- lation is described in quantitative impact study 5. We describe a general approach to risk measurement and we show some particular practical measures used to risk measurement. We explain under what conditions the standard formula or its parts can be replaced by internal model. Next, we show disadvantages of using the stan- dard formula and we propose possible internal model to calculate risk premiums and risk reserves in non-life insurance. Finally we apply the proposed model for calculation risk reverses in non-life insurance in practice. Keywords: Standard formula, Risk measurement, Solvency II, Internal model;
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Logistic regression with applications in financial sector
Bílková, Kristýna ; Branda, Martin (advisor) ; Pešta, Michal (referee)
In this bachelor thesis binary logistic regression model is described. Its parameters are estimated by maximum likelihood method. Newton-Raphson's algorithm is used for enumeration of these estimates. There are defined some statistics for testing the significance of the coefficients. Then stepwise regression is desribed. For assessing the quality of the model Pearson's Chi Square Test and Hosmer-Lemeshow's Test of the goodness of fit are defined. Diversification abilitz of the model is illustrated bz the Loreny curve and is quantificated by Gini coefficient, Kolmogorov-Smirnov statistics and generalized coefficient of determination. The theoretical knowledge is applied to insurance area data.
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