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Quantifying Mortality and Longevity Risk by Means of Stochastic Models
Plotnikova, Valeriya ; Mazurová, Lucie (advisor) ; Kříž, Pavel (referee)
In this thesis we investigate the structure of the generalized age-period-cohort mortality model and we comment on the key components of its structure. As an example of the generalized age- period-cohort model we take a closer look at the widely used Lee-Carter mortality model. We further construct mortality models for the Czech male and female populations, by using a certain procedure that involves expert judgment. To project mortality rates we choose the most suitable time series processes for the selected parameters in the model. Finally, we describe and implement the value at risk framework for the longevity risk, which is one of the possible applications where the obtained mortality models can be used in practice. In particular, we investigate how much a temporary life annuity liability might change based on new information over the course of one year.
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Strong stationary times and convergence of Markov chains
Suk, Luboš ; Prokešová, Michaela (advisor) ; Kříž, Pavel (referee)
In this thesis we study the estimation of speed of convergence of Markov chains to their stacionary distributions. For that purpose we will use the method of strong stationary times. We focus on irreducible and aperiodic chains only since in that case the existence of exactly one stationary distribution is guaranteed. We introduce the mixing time for a Markov chain as the time needed for the marginal distribution of the chain to be sufficiently close to the stationary dis- tribution. The distance between two distributions is measured by the total variation distance. The main goal of this thesis is to construct an appropriate strong stationary time for selected chains and then use it for obtaining an upper bound for the mixing time.
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The identification function for the convergence in probability with an application in the estimation theory
Kříž, Pavel ; Štěpán, Josef (advisor) ; Hlubinka, Daniel (referee)
In the present work we introduce the concept of probability limit identification function (PLIF) as it is done in [6]. This function identifies almost surely the value of the probability limit of a sequence of random variables on the basis of one realization of the sequence. According to the same article we show the construction of PLIF for real valued random variables from the special PLIF for 0-1 valued random variables. Following the method described in [8] we prove the existence of the universal PLIF for real valued random variables under the continuum hypothesis. Next we show that there are no borel measurable special PLIFs for 0-1 valued random variables (as well as PLIFs for real valued random variables). We use the proof that is published in [2]. Then we extend the construction of PLIF from R to any separable metrizable topological space. This PLIF may be used e.g. for creating functional representations of stochastic integrals and weak solutions of stochastic differential equations.
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Kolmogorov-Chentsov Theorem
Lebeda, Matěj ; Čoupek, Petr (advisor) ; Kříž, Pavel (referee)
Is there a sufficient condition for continuity of sample paths of a random process? Or, is it at least possible to modify the process so that the paths would already be continuous? An affirmative answer is given by the Kolmogorov- Chentsov theorem, whose statement and proof are the subject of this thesis. First, we introduce the notion of a random process and briefly focus on the so-called Gaussian processes. The main focus of the second chapter is the Kolmogorov- Chentsov theorem, its proof and some auxiliary assertions are given. In the final third chapter, we deal with the applications of the theorem to some well-known Gaussian processes such as the Wiener process or the Brownian bridge. Finally, we look into the Poisson process, which on the contrary does not satisfy the condition of the theorem. 1
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Efficiency of bonus-malus systems
Hrbáčová, Daniela ; Mazurová, Lucie (advisor) ; Kříž, Pavel (referee)
This thesis deals with bonus-malus systems during motor third party liability insu- rance. Bonus-malus system is used to adjust apriori set tariff premium according to the individual claims. This fruther adjustment of premium rate serves to make claim costs even more fair distributed among policyholders in one tariff class. We model driver's passage through the system as a homogenous Markov chain. The result of this thesis is the assessment of the efficiency of the systems used by two insurance companies in the Czech Republic on a model insurance portfolio. 1
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Selected properties of bivariate and multivariate random walks.
Nguyen, Huy Quang ; Hlubinka, Daniel (advisor) ; Kříž, Pavel (referee)
This thesis deals with random walks with emphasis on multivariate random walks. We focus mainly on return of the random walk to the origin in two dimensions. Some results are generalized in any dimension. Specifically we discuss the probability of return to the origin, probability of the first return to the origin and expected time of the first return. In the thesis we also find the arcsine laws and short simulation study focused on multivariate version of this topic. 1
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System to support decision making in dental implantology - quality of life of patients with dental implants
Kříž, Pavel ; Dostálová, Taťjana (advisor) ; Mazánek, Jiří (referee) ; Zvárová, Jana (referee)
Title: System to support decission making in dental implantology - quality of life of patients with dental implants Author: MUDr. Pavel Kříž Department: Paediatric Stomatology of 2nd Medical School and Faculty Hospital Motol, V Úvalu 84, 150 06 Prague 5 Supervisor: Prof. MUDr. Taťjana Dostálová, DrSc., MBA Supervisor's e-mail: tatjana.dostalova@fnmotol.cz Dental implants are the method of choice in the treatment of missing tooth/teeth replacement. Implant therapy must be preceded by a detailed examination and the overall treatment plan. As an aid for decision-making for dentists was created decision-making scheme, which gradually, logically and schematically guides the dentists in this particular situation. Health is closely related to the quality of life. Our work evaluates the oral health-related quality of life (OHRQoL) of patients with dental implants. The aim of our study was to determine whether treatment with a dental implant(s) improve(s) OHRQoL. We created a questionnaire to determine the quality of life before and after implantation. In our study, we evaluated only patients who were treated by the only one implantological system to eliminate the influence of other systems on the quality of the results of the study; we evaluated a total of 297 implants. It was assessed a total of 97...
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Parameter estimation for Ornstein-Uhlenbeck process
Martinková, Sandra ; Kříž, Pavel (advisor) ; Maslowski, Bohdan (referee)
The Ornstein-Uhlenbeck process has countless practical applications most of which rely on having previously estimated the drift parameter. The literature offers two basic estima- tors - the least-squares estimator, which coincides with the maximum-likelihood estimator for Ornstein-Uhlenbeck process, and the method-of-moments estimator. However, the sim- ilarity in asymptotic properties of these estimators means that choosing which one to use is more of a random guess than an educated decision. This thesis focuses on finding dif- ferences between the two estimators when applied to the Ornstein-Uhlenbeck trajectories generated in R. The simulation study performed suggests that the method-of-moments is better suited when the initial condition is close to zero even if the observations are col- lected sparsely. On the other hand, the precision of the least-squares estimator is better when the initial condition is further away from zero, but it still requires having dense data points. Under the conditions of this study, the least-squares estimator performs better compared to the method-of-moments if the absolute value of the initial condition is large. On the other hand, the method-of-moments is superior in cases where we have infrequent observations and long time horizon.
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The Study of Product Lifecycle Management in an Enterprise
Kříž, Pavel ; Gabrhelíková, Zdeňka (referee) ; Jurová, Marie (advisor)
The thesis explores sustainable development of a company focusing on PLM system in the environment of an industrial company. In the theoretical part the lifecycle of a product (PLM) is described as well as its demands and benefits focusing on the area of mechatronics. The following analysis of current IT environment gives information which is used as the basis for adjustments of the current system in a way to eliminate narrow places of the slim production of the company, avoid wasting and achieve sustainable development of the company while using current technologies.
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