
Claims reserve volatility and bootstrap with aplication on historical data with trend in claims development
Malíková, Kateřina ; Pešta, Michal (advisor) ; Zichová, Jitka (referee)
This thesis deals with the application of stochastic claims reserving methods to given data with some trends in claims development. It describes the chain ladder method and the generalized linear models as its stochastic framework. Some simple functions are suggested for smoothing the origin and development period coefficients from the estimated model. The extrapolation is also considered for estimation of the unobserved tail values. The residual bootstrap is used for the reparameterized model in order to get the predictive distribution of the estimated reserve together with its standard deviation as a measure of volatility. Solvency capital requirement in one year time horizon is also calculated. 1


Estimations of risk with respect to monthly horizon based on the twoyear time series
Myšičková, Ivana ; Houfková, Lucia (advisor) ; Pešta, Michal (referee)
The thesis describes commonly used measures of risk, such as volatility, Value at Risk (VaR) and Expected Shortfall (ES), and is tasked with creating models for measuring market risk. It is concerned with the risk over daily and over monthly horizons and shows the shortcomings of a squarerootoftime approach for converting VaR and ES between horizons. Parametric models, geometric Brownian motion (GBM) and GARCH process, and nonparametric models, historical simulation (HS) and some its possible improvements, are presented. The application of these mentioned models is demonstrated using real data. The accuracy of VaR models is proved through backtesting and the results are discussed. Part of this thesis is also a simulation study, which reveals the precision of VaR and ES estimates.


Tests for Combination of Correlation Coefficients
Kulíšková, Michaela ; Hudecová, Šárka (advisor) ; Pešta, Michal (referee)
This bachelor thesis is focused on tests for correlation coefficients. Fundamental knowledge about correlation coefficient are reminded as well as tests for correlation coefficient based on estimated correlation coefficient. Then three main methods for combining more correlation coefficients  Fisher`s method, Linear combination test with Ztransformation and Hotelling transformation  are described, simulated and compared. These tests have several assumptions such as that k correlation coefficients are known as well as the range of random samples from which they were calculated plus it is assumed that these correlation coefficients are equal.


ksample problem with ordered alternative
Nováková, Martina ; Hlávka, Zdeněk (advisor) ; Pešta, Michal (referee)
In this thesis we deal with ksample problem with ordered alternative. At the beginning of the thesis isotonic regression is introduced. We use isotonic regression for maximum likelihood estimation of ordered parameters. In the second chapter, we describe the χ2 and E 2 tests that use the knowledge of isotonic regression and are based on the likelihood ratio. The exact null hypothesis distributions of their test statistics are derived in detail. The onesided studentized range test is also further described. At the end of the thesis, we show the use of the E 2 test on the real data. 1


Loss reserving for individual claimbyclaim data
Bednárik, Vojtěch ; Pešta, Michal (advisor)
This thesis covers stochastic claims reserving in nonlife insurance based on individual claims developments. Summarized theoretical methods are applied on data from Czech Insurers' Bureau created for educational purposes. The problem of estimation is divided into four parts: occurence process generating claims, delay of notification, times between events and payments. Each part is estimated separately based on maximum likelihood theory and final estimates allow us to obtain an estimate of future liabilities distribution. The results are very promising and we believe this method is worth of a further research. Contribution of this work is more rigorous theoretical part and application on data from the Czech market with some new ideas in practical part and simulation. 1


Nonhomogeneous Poisson process  estimation and simulation
Vedyushenko, Anna ; Pešta, Michal (advisor) ; Pawlas, Zbyněk (referee)
This thesis covers nonhomogeneous Poisson processes along with estimation of the intensity (rate) function and some selected simulation methods. In Chapter 1 the main properties of a nonhomogeneous Poisson process are summarized. The main focus of Chapter 2 is the general maximum likelihood estimation procedure adjusted to a nonhomogeneous Poisson process, together with some recommen dations about calculation of the initial estimates of the intensity function param eters. In Chapter 3 some general simulation methods as well as the methods designed specially for log linear and log quadratic rate functions are discussed. Chapter 4 contains the application of the described estimation and simulation methods on real data from nonlife insurance. Furthermore, the considered sim ulation methods are compared with respect to their time efficiency and accuracy of the simulations. 1


Analysis of Variance with Random Effects
Hamerníková, Iva ; Komárek, Arnošt (advisor) ; Pešta, Michal (referee)
The aim of this thesis is to describe and derive the test of analysis of variance with random effects. At first we introduce a summary of results from the theory of probability which will be important in future derivations. Then we define the oneway classification model with fixed effects and propose the test statistics to test the equality of group means. In the following part we define the oneway classification model with random effects and derive properties of observations in this model. Under the assumption of balanced data we define sums of squares and derive their properties, which allow us to use them to create the test statistic. Finally we will use simulations in R to verify whether the ANOVA test with random effects observes the significance level when normality assumptions are violated.


Truncated data and stochastic claims reserving
Marko, Dominik ; Pešta, Michal (advisor) ; Mazurová, Lucie (referee)
In this thesis stochastic claims reserving under the model of randomly trun cated data is presented. For modelling the claims, a compound Poisson process is assumed. Introducing a random variable representing the delay between oc currence and reporting of a claim, a probability model of IBNR claims is built. The fact that some claims are incurred but not reported yet leads to truncated data. Basic results of nonparametric statistical estimation under the model of randomly truncated data are shown, which can be used to obtain an estimate of IBNR claims reserves. Theoretical background is then used for application on real data from Czech Insurers' Bureau. 36


Structural Equation Models with Application in Social Sciences
Veselý, Václav ; Pešta, Michal (advisor) ; Maciak, Matúš (referee)
We investigate possible usage of ErrorsinVariables estimator (EIV), when esti mating structural equations models (SEM). Structural equations modelling pro vides framework for analysing complex relations among set of random variables where for example the response variable in one equation plays role of the predic tor in another equation. First an overview of SEM and some common covariance based estimators is provided. Special case of linear regression model is investi gated, showing that the covariance based estimators yield the same results as ordinary least squares. A compact review of EIV models follows, ErrorsinVariables models are re gression models where not only response but also predictors are assumed to be measured with an error. Main contribution of this paper then lies in defining modifications of the EIV estimator to fit in the SEM framework. General opti mization problem to estimate the parameters of structural equations model with errorsinvariables si postulated. Several modifications of two stage least squares are also proposed for future research. Equationwise ErrorsinVariables estimator is proposed to estimate the coeffi cients of structural equations model. The coefficients of every structural equation are estimated separately using EIV estimator. Some theoretical conditions...


Statistical inference in varying coefficient models
Splítek, Martin ; Maciak, Matúš (advisor) ; Pešta, Michal (referee)
Tato práce se zabývá modely s promìnlivými koe cienty se za mìøením na statistickou inferenci. Hlavní my¹lenkou tìchto modelù je pou¾ití regresních koe cientù, mìnících se v závislosti na nìjakém modi kátoru vlivu, namísto konstantních koe cientù klasické lineární regrese. Nejprve si de nujeme tyto modely a jejich odhadové procedury, kterých bylo doposud publikováno nì kolik variant. K odhadu se pou¾ívá lokální regrese nebo rùzné druhy splajnù { vyhlazovací, polynomiální èi penalizované. Od metody odhadu se následnì od víjí i daná statistická inference, ke které uvedeme odvozené vychýlení, rozptyl, asymptotickou normalitu, kon denèní pásma a testování hypotéz. Hlavním cílem na¹í práce je kompaktnì shrnout vybrané metody a jejich inferenci. Na závìr je navr¾ena proceduru pro výbìr promìnných.
