
The power of two sample tests
Rózsahegyi, Dominik ; Maciak, Matúš (advisor) ; Nagy, Stanislav (referee)
Twosample tests are commonly used in practice, for example in scientific sphere or financial sectors. The power of test is also an important feature and express probability that the test will reject invalid null hypothesis. In this work we introduce four basic tests in which we compare some parameters of two popu lations. The reader gets to know with basic terms of hypothesis testing which are necessary for introduction of tests. For each test we use simulation for estima tion of the power and observe its behavior with different distributions of samples, ranges or selected null and alternative hypothesis. Based on obtained results we compare chosen tests and discuss suitability for using them in different cases. 1


Tests for the Poisson distribution
Trusina, Filip ; Pawlas, Zbyněk (advisor) ; Nagy, Stanislav (referee)
In this work we deal with the question whether a sequence of independent identically distributed random variables comes from the Poisson distribution. For this task we present two different approaches and couple of tests for each appro ach. The first approach is based on the asymptotic approximation of distribution of test statistics. The second approach uses generation of test samples. Based on simulations done by us, we discuss the power of individual tests and their advantages and disadvantages. 1


Edgeworth expansion
Dzurilla, Matúš ; Omelka, Marek (advisor) ; Nagy, Stanislav (referee)
This thesis is focused around Edgeworth's expansion for approximation of distribution for parameter estimation. Aim of the thesis is to introduce term Edgeworth's expansion, its assumptions and terminology associated with it. Afterwards demonstrate process of deducting first term of Edgeworth's expansion. In the end demonstrate this deduction on examples and compare it with different approximations (mainly central limit theorem), and show strong and weak points of Edgeworth's expansion.


Halfspace median
Říha, Adam ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
In this thesis we introduce the halfspace median, which is one of the possibilities how to extend the classical median from a onedimensional space to spaces with several dimensions. Firstly we deal with the halfspace depth, which is a function that assigns to each point the minimum probability of a halfspace that contains it. Then we define the halfspace median and show its existence. Partially, we also deal with special types of symmetry measures for convex sets and random vectors and what follows from them, such as when the median and the center of symmetry are the same point. We also study the boundaries that, under certain assumptions, enclose the depth. We state sufficient conditions for acquiring the halfspace median, which are determined by the socalled ray basis theorem. Finally we look at the similarities of this topic with convex geometry.


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


Boxplot for multivariate data
Brabenec, Tomáš ; Nagy, Stanislav (advisor) ; Hlávka, Zdeněk (referee)
We will introduce three methods of extension of the classical Tukey's Boxplot for multivariate data. These are the Rangefinder, the Relplot and the Bagplot. To implement the methods, we will need the notions like Mahalanobis distance, elliptically symmetric distributions and halfspace depth. A big part of the thesis is focused on the construction of the Relplot and the Bagplot. We will also discuss, how do these methods detect outliers and what are their advantages and disadvantages. This work contains many examples and illustrating images. 1


Geometric approach to the estimation of scatter
Bodík, Juraj ; Nagy, Stanislav (advisor) ; Antoch, Jaromír (referee)
In this thesis we describe improved methods of estimating mean and scatter from multivariate data. As we know, the sample mean and the sample variance matrix are nonrobust estimators, which means that even a small amount of measurement errors can seriously affect the resulting estimate. We can deal with that problem using MCD estimator (minimum covariance determinant), that finds a sample variance matrix only from a selection of data, specifically those with the smallest determinant of this matrix. This estimator can be also very helpful in outlier detection, which is used in many applications. Moreover, we will introduce the MVE estimator (minimum volume ellipsoid). We will discuss some of the properties and compare these two estimators.


Edgeworth expansion
Dzurilla, Matúš ; Omelka, Marek (advisor) ; Nagy, Stanislav (referee)
This thesis is focused around Edgeworths expansion for aproximation of distribution for parameter estimation. Aim of the thesis is to introduce term Edgeworths expansion, its assumptions and terminology associeted with it. Afterwords demonstrate process of deducting first term of Edgeworths expansion. In the end demonstrate this deduction on examples and compare it with different approximations (mainly central limit theorem), and show strong and weak points of Edgeworths expansion.


Principal components
Zavadilová, Anna ; Hlávka, Zdeněk (advisor) ; Nagy, Stanislav (referee)
This thesis presents principal components as a useful tool for data dimensio nality reduction. In the first part, the basic terminology and theoretical properties of principal components are described and a biplot construction is derived there as well. Besides, heuristic methods for a choice of the optimum number of prin cipal components are summarised there. Subsequently, asymptotical properties of sample eigenvalues of covariance and white Wishart matrices are described and cases of equality of some eigenvalues are distinguished at the same time. In the second part of the thesis, asymptotic distribution of the largest eigenva lue of white Wishart matrices is described, completed with graphic illustrations. A test of the number of significant eigenvalues is suggested on the basis of this limiting distribution, and the connection of this test to the number of suitable principal components is presented. The final part of the thesis provides an over view of advanced computational methods for the choice of an adequate number of principal components. The thesis is completed with graphical illustrations and a simulation study using Wolfram Mathematica and R.


Calibration Estimators in Survey Sampling
Klička, Petr ; Omelka, Marek (advisor) ; Nagy, Stanislav (referee)
V této práci se zabýváme odhady populačního úhrnu s využitím pomoc ných informací. V práci je popsán obecný regresní odhad a předpoklady, za kterých je splněna asymptotická normalita tohoto odhadu. Dále jsou zde po psány kalibrační odhady a zmínka o jejich asymptotické ekvivalenci s obec ným regresním odhadem. Odvozené závěry aplikujeme na data z RADIO PROJEKTu a porovnáme je s výsledky získanými společnostmi, které tento projekt realizovali. Na závěr pomocí simulací porovnáme skutečné pravdě podobnosti pokrytí interval· spolehlivosti pro populační úhrn spočítané na základě teorie uvedené v této práci a na základě metod společností realizu jících RADIOPROJEKT. 1
