|
|
Fisherovo-Binghamovo rozdělení
Malá, Olivia Caroline ; Hlávka, Zdeněk (advisor) ; Hlubinka, Daniel (referee)
This thesis is an introduction into directional statistics, a subdiscipline of statistics that occupies itself with directional data. Because of the special structure of the sample spaces, which are n-dimensional hyperspheres, the statis- tical theory has to be adjusted. We start on the circle, where we define the circular random variable (also called random angle) together with its characterizations, and continue with studying estimators of its parameters. Subsequently, we generalize the results to the n- dimensional case. Further follows an overview of the Fisher-Bingham family of probability distributions with a more detailed presentation of the von Mises dis- tribution as a representative of the family on the circle. 1
|
|
| |
|
| |
|
|
Level Sets of Multivariate Density Functions and their Estimates
Kubetta, Adam ; Hlubinka, Daniel (advisor) ; Zichová, Jitka (referee)
A level set of a function is defined as the region, where the function gets over the specified level. A level set of the probability density function can be considered an alternative to the traditional confidence region because on certain conditions the level set covers the region with minimal volume over all regions with a given confidence level. The benefits of using level sets arise in situations where, for example, the given random variables are multimodal or the given random vectors have strongly correlated components. This thesis describes estimates of the level set by means of a so called plug-in method, which first estimates density from the data set and then specifies the level set from the estimated density. In addition, explicit direct methods are also studied, such as algorithms based on support vectors or dyadic decision trees. Special attention is paid to the nonparametric probability density estimates, which form an essential tool for plug-in estimates. Namely, the second chapter describes histograms, averaged shifted histograms, kernel density estimates and its generalization. A new technique transforming kernel supports is proposed to avoid the so called boundary effect in multidimensional data domains. Ultimately, all methods are implemented in Mathematica and compared on financial data sets.
|
|
|
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.
|
|
| |
|
|
Maximum likelihood methods; selected problems
Chlubnová, Tereza ; Hlubinka, Daniel (advisor) ; Hlávka, Zdeněk (referee)
Maximum likelihood estimation is one of statistical methods for estimating an unknown parameter. It is often used because of a simple calculation of the estimator and also for characteristics of this estimator, which the method provides under some conditions. In the thesis we prove a consistence of the estimator under conditions of regularity and uniqueness of the root of the likelihood equation. If we add other assumptions we show its asymptotic normality and we expand this result from the one-dimensional parameter to the multi-dimensional parameter. The main result of the thesis lies in exercises, in which we cannot express the maximum likelihood estimator in general, but we can show its existence, uniqueness and asymptotic normality. Moreover we demonstrate the utilization of asymptotic normality of the estimator for asymptotic hypothesis tests and confidence intervals of the parameter. Powered by TCPDF (www.tcpdf.org)
|
|
|
Quantification of multivariate risk
Hilbert, Hynek ; Hlubinka, Daniel (advisor) ; Hudecová, Šárka (referee)
In the present work we study multivariate extreme value theory. Our main focus is on exceedances over linear thresholds. Smaller part is devoted to exce- edances over elliptical thresholds. We consider extreme values as those which belong to remote regions and investigate convergence of their distribution to the limit distribution. The regions are either halfspaces or ellipsoids. Working with halfspaces we distinguish between two setups: we either assume that the distribution of extreme values is directionally homogeneous and we let the halfspaces diverge in any direction, or we assume that there are some irre- gularities in the sample cloud which show us the fixed direction we should let the halfspaces drift out. In the first case there are three limit laws. The domains of attraction contain unimodal and rotund-exponential distributions. In the second case there exist a lot of limit laws without general form. The domains of attraction also fail to have common structure. The similar situation occurs for the exceedances over elliptical thresholds. The task here is to investigate convergence of the random vectors living in the complements of ellipsoids. For all, the limit distributions are determined by affine transformations and distribution of spectral measure. 1
|
|
|
Stochastic Evolution Equations
Čoupek, Petr ; Maslowski, Bohdan (advisor) ; Garrido-Atienza, María J. (referee) ; Hlubinka, Daniel (referee)
Stochastic Evolution Equations Petr Čoupek Doctoral Thesis Abstract Linear stochastic evolution equations with additive regular Volterra noise are studied in the thesis. Regular Volterra processes need not be Gaussian, Markov or semimartingales, but they admit a certain covariance structure instead. Particular examples cover the fractional Brownian motion of H > 1/2 and, in the non-Gaussian case, the Rosenblatt process. The solution is considered in the mild form, which is given by the variation of constants formula, and takes values either in a separable Hilbert space or the space Lp(D, µ) for large p. In the Hilbert-space setting, existence, space-time regularity and large-time behaviour of the solutions are studied. In the Lp setting, existence and regularity is studied, and in concrete cases of stochastic partial differential equations, the solution is shown to be a space-time continuous random field.
|
|
| |
guest ::
RSS feed.