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Chaotic random variables in applied probability
Večeřa, Jakub ; Beneš, Viktor (advisor) ; Reitzner, Matthias (referee) ; Pawlas, Zbyněk (referee)
This thesis deals with modeling of particle processes. In the first part we ex- amine Gibbs facet process on a bounded window with discrete orientation distri- bution and we derive central limit theorem (CLT) for U-statistics of facet process with increasing intensity. We calculate all asymptotic joint moments for interac- tion U-statistics and use the method of moments for deriving the CLT. Moreover we present an alternative proof which makes use of the CLT for U-statistics of a Poisson facet process. In the second part we model planar segment processes given by a density with respect to the Poisson process. Parametric models involve reference distributions of directions and/or lengths of segments. Statistical methods are presented which first estimate scalar parameters by known approaches and then the reference distribution is estimated non-parametrically. We also introduce the Takacs-Fiksel estimate and demonstrate the use of estimators in a simulation study and also using data from actin fibres from stem cells images. In the third part we study a stationary Gibbs particle process with determin- istically bounded particles on Euclidean space defined in terms of a finite range potential and an activity parameter. For small activity parameters, we prove the CLT for certain statistics of this...
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Sample Quantiles
Hrušková, Iveta ; Komárek, Arnošt (advisor) ; Nagy, Stanislav (referee)
If the distribution of random variable is uknown, we are not able to figure out the value of theoretical quantile. In case there is a random sample from this distribution, it is possible to estimate the value of theoretical quantile. This es- timation is called sample quantile. This work is focused on nine frequently used varieties of sample quantile. They will be compared by means of characteristics that can be examined when speaking about sample quantile. All these varieties will be demonstrated on simple example. Finally, there will be shown that all these versions of sample quantile are consistent estimators of theoretical quan- tile. The construction of confidence interval for theoretical quantile will be the topic of the final part of the work. 1
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Non-homogeneous Poisson process - estimation and simulation
Vedyushenko, Anna ; Pešta, Michal (advisor) ; Pawlas, Zbyněk (referee)
This thesis covers non-homogeneous Poisson processes along with estimation of the intensity (rate) function and some selected simulation methods. In Chapter 1 the main properties of a non-homogeneous Poisson process are summarized. The main focus of Chapter 2 is the general maximum likelihood estimation procedure adjusted to a non-homogeneous 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 non-life insurance. Furthermore, the considered sim- ulation methods are compared with respect to their time efficiency and accuracy of the simulations. 1
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Measurement and forecasting of tax revenues
Bayer, Ondřej ; Vítek, Leoš (advisor) ; Pavel, Jan (referee)
This work is aimed on measurement and forecasting of tax revenues. The target is to explain issue of tax revenues measurement by other ways and after that to create forecasting models of tax revenues since 2001 to 2011. For achieving this target is used literature and economethric modeling methodology. At first is analyzed difference between acrual and cash methods of tax revenue's measurement from the view of different methodologies. After that econometric models are created for each of the choosen tax revenues. These models are analyzed for the quality of forecasting. From the results is clear, that tax revenues have a part, which could be described by macroeconomic variables. But these models reveals, that part of the tax revenue could be described by autonomous or random component. Models themselves could be affected by failure of elemental econometrics assumptions.
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