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Truncated data
Redek, David ; Pešta, Michal (advisor) ; Vávra, Jan (referee)
This thesis deals with truncated distributions. Firstly, the case of the truncated normal distribution is analyzed in detail. Two estimation methods are described - method of moments and maximum likelihood - together with the discussion of their properties and confidence region construction. A more advanced method - Bayes estimator - is briefly presented. Secondly, the truncated gamma distribution is analyzed, however, in less detail than the normal case. The theoretical part is closed with a method for estimating truncation boundaries when not even those are known. Throughout the thesis, results from multiple articles by various authors are summarized and presented in a unified notation. The numerical part deals with the analysis of a real dataset, describing the height of soldiers in the U.S. Army. The built theory is transformed into R code and executed, indicating the correctness of our theoretical results. 1
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Modern Asymptotic Perspectives on Errors-in-variables Modeling
Pešta, Michal
Charles University in Prague Faculty of Mathematics and Physics ABSTRACT OF DOCTORAL THESIS Michal Pešta MODERN ASYMPTOTIC PERSPECTIVES ON ERRORS-IN-VARIABLES MODELING A linear regression model, where covariates and a response are subject to errors, is considered in this thesis. For so-called errors-in-variables (EIV) model, suitable error structures are proposed, various unknown parameter estimation techniques are performed, and recent algebraic and statistical results are summarized. An extension of the total least squares (TLS) estimate in the EIV model-the EIV estimate-is in- vented. Its invariant (with respect to scale) and equivariant (with respect to the covariates' rotation, to the change of covariates direction, and to the interchange of covariates) properties are derived. Moreover, it is shown that the EIV estimate coincides with any unitarily invariant penalizing solution to the EIV problem. It is demonstrated that the asymptotic normality of the EIV estimate is computationally useless for a construction of confidence intervals or hypothesis testing. A proper bootstrap procedure is constructed to overcome such an issue. The validity of the bootstrap technique is proved. A simulation study and a real data example assure of its appropriateness. Strong and uniformly strong mixing errors are taken...
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Modelování velkých škod
Petrová, Barbora ; Pešta, Michal (advisor)
Title: Large claims modeling Author: Barbora Zuzáková Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michal Pešta, Ph.D. Abstract: This thesis discusses a statistical modeling approach based on the extreme value theory to describe the behaviour of large claims of an insurance portfolio. We focus on threshold models which analyze exceedances of a high threshold. This approach has gained in popularity in recent years, as compared with the much older methods based directly on the extreme value distributions. The method is illustated using the group medical claims database recorded over the periods 1997, 1998 and 1999 maintained by the Society of Actuaries. We aim to demonstrate that the proposed model outperforms classical parametric distri- butions and thus enables to estimate high quantiles or the probable maximum loss more precisely. Keywords: threshold models, generalized Pareto distribution, large claims. 1
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Estimations of risk with respect to monthly horizon based on the two-year 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 square-root-of-time approach for converting VaR and ES between horizons. Parametric models, geometric Brownian motion (GBM) and GARCH process, and non-parametric 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.
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Alternative approach to BEL calculations for life insurance
Teichmannová, Zuzana ; Pešta, Michal (advisor) ; Mazurová, Lucie (referee)
This thesis presents an alternative approach for the Best Estimate of Liabilities (BEL) approximation in life insurance. The work summarizes the basic theoretical knowledge about reserving in life insurance and deterministic or stochastic projection of future cash flows which is a method commonly used to model the value of BEL. This thesis also presents the theory about durations. We use partial key rate durations to approximate the value of BEL. The proposed approach is tested on a real example life insurance product with profit share. The resulting approximations are close to real values and when partial durations obtained by deterministic calculations are used, the preparation of the approximation is not computationally demanding. 1
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Analysis of several acceleration techniques for life insurance liability value determination
Drahokoupil, Matěj ; Pešta, Michal (advisor) ; Branda, Martin (referee)
The aim of the diploma thesis is to apprise the reader with a basic life insur- ance projection method which is used for the valuation of insurance company's liabilities. The basic projection method can be extremely time consuming in practise so another two variance reduction methods and their combination are presented to obtain either more precise liabilities estimation, or to reduce the time required for the projection. The presented methods are antithetic variate method, control-variate method and their combination later called integrated control-variate method. The final outcome of the thesis is simulation experi- ment which evaluates the liabilities of the group of policies and comparison of the presented variance reduction methods. 1
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Multivariate generalized autoregressive conditional heteroscedasticity models
Nováková, Martina ; Pešta, Michal (advisor) ; Maciak, Matúš (referee)
This master thesis deals with extension of the univariate GARCH model to multivari- ate models. We present individual models and deal with methods of their estimation. Then we describe some statistical tests for diagnosting the models. We have programmed in the statistical software R one of them - the Ling-Li test. Afterwards we apply selected models to real data of stock market index S&P 500, stock market index Russell 2000 and stocks of crude oil. For the GO-GARCH model, we compare all available estimation methods and show their differences. Then we compare the results of all models with each other and also with univariate models in terms of estimates of conditional variances, estimates of conditional correlations and also in terms of computational complexity. 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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Testing equality of means by confidence intervals
Jandl, Vojtěch ; Kulich, Michal (advisor) ; Pešta, Michal (referee)
We deal with testing the equality of means using confidence intervals. Firstly, we introduce the methods of testing that have already been published. The advantage of these methods is that one can present the underlying confidence intervals alongside the result of the test without doing further calculations. In the second part we discuss the necessary assumptions and by that we extend the Noguchi's method to discrete distribu- tions. Also, we derive a generalization of the Noguchi's method for testing the equality of other parameters than means, based on the assumption of asymptotic normality of their consistent estimates. Lastly, we conduct a simulation study in order to compare the methods we discussed. We found out that the Noguchi's method is a worthy alternative to the often-used Welch test bearing the advantage of being able to present extra visual output in the form of the underlying confidence intervals. In comparison to other methods the Noguchi's method yields better results in the case of unequal or small sample sizes. Unlike other methods it can also be used for testing in the paired sample case. 1
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Various change point estimation methods
Šimonová, Soňa ; Pešta, Michal (advisor) ; Hušková, Marie (referee)
This thesis aims to give a comprehensive account of some of the most recent methods of a change point estimation. The literature on the change point estimation shows a variety of approaches to deal with this subject. Among them, tests based on the popular CUSUM process, likelihood ratio tests, wild binary segmentation and some of the most recent techniques on the change point estimation in panel data are all covered by this paper. The case of dependent panels is discussed as well. The practical part of the study is focused on application of the wild binary segmentation method on weekly log-returns of the Dow Jones stock index. Firstly, we fit a GARCH model to the analysed time series. We next use the wild binary segmenatation method to detect structural changes in the mean of the original time series. Next, we apply the same method to the residuals from the GARCH fit. We analyse several penalization criteria proposed by previous studies and evaluate their effects on the estimated number and locations of the change points in the given data set. 1
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