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Weighted Halfspace Depths and Their Properties
Kotík, Lukáš
Statistical depth functions became well known nonparametric tool of multivariate data analyses. The most known depth functions include the halfspace depth. Although the halfspace depth has many desirable properties, some of its properties may lead to biased and misleading results especially when data are not elliptically symmetric. The thesis introduces 2 new classes of the depth functions. Both classes generalize the halfspace depth. They keep some of its properties and since they more respect the geometric structure of data they usually lead to better results when we deal with non-elliptically symmetric, multimodal or mixed distributions. The idea presented in the thesis is based on replacing the indicator of a halfspace by more general weight function. This provides us with a continuum, especially if conic-section weight functions are used, between a local view of data (e.g. kernel density estimate) and a global view of data as is e.g. provided by the halfspace depth. The rate of localization is determined by the choice of the weight functions and theirs parameters. Properties including the uniform strong consistency of the proposed depth functions are proved in the thesis. Limit distribution is also discussed together with some other data depth related topics (regression depth, functional data depth)...
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Additive combinatorics and number theory
Hančl, Jaroslav ; Klazar, Martin (advisor)
We present several results for growth functions of ideals of different com- binatorial structures. An ideal is a set downward closed under a containment relation, like the relation of subpartition for partitions, or the relation of induced subgraph for graphs etc. Its growth function (GF) counts elements of given size. For partition ideals we establish an asymptotics for GF of ideals that do not use parts from a finite set S and use this to construct ideal with highly oscillating GF. Then we present application characterising GF of particular partition ideals. We generalize ideals of ordered graphs to ordered uniform hypergraphs and show two dichotomies for their GF. The first result is a constant to linear jump for k-uniform hypergraphs. The second result establishes the polynomial to exponential jump for 3-uniform hypergraphs. That is, there are no ordered hypergraph ideals with GF strictly inside the constant-linear and polynomial- exponential range. We obtain in both dichotomies tight upper bounds. Finally, in a quite general setting we present several methods how to generate for various combinatorial structures pairs of sets defining two ideals with iden- tical GF. We call these pairs Wilf equivalent pairs and use the automorphism method and the replacement method to obtain such pairs. 1
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Additive combinatorics and number theory
Hančl, Jaroslav ; Klazar, Martin (advisor)
We present several results for growth functions of ideals of different com- binatorial structures. An ideal is a set downward closed under a containment relation, like the relation of subpartition for partitions, or the relation of induced subgraph for graphs etc. Its growth function (GF) counts elements of given size. For partition ideals we establish an asymptotics for GF of ideals that do not use parts from a finite set S and use this to construct ideal with highly oscillating GF. Then we present application characterising GF of particular partition ideals. We generalize ideals of ordered graphs to ordered uniform hypergraphs and show two dichotomies for their GF. The first result is a constant to linear jump for k-uniform hypergraphs. The second result establishes the polynomial to exponential jump for 3-uniform hypergraphs. That is, there are no ordered hypergraph ideals with GF strictly inside the constant-linear and polynomial- exponential range. We obtain in both dichotomies tight upper bounds. Finally, in a quite general setting we present several methods how to generate for various combinatorial structures pairs of sets defining two ideals with iden- tical GF. We call these pairs Wilf equivalent pairs and use the automorphism method and the replacement method to obtain such pairs. 1
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Additive combinatorics and number theory
Hančl, Jaroslav ; Klazar, Martin (advisor) ; Balogh, Jozsef (referee) ; Nedela, Roman (referee)
We present several results for growth functions of ideals of different com- binatorial structures. An ideal is a set downward closed under a containment relation, like the relation of subpartition for partitions, or the relation of induced subgraph for graphs etc. Its growth function (GF) counts elements of given size. For partition ideals we establish an asymptotics for GF of ideals that do not use parts from a finite set S and use this to construct ideal with highly oscillating GF. Then we present application characterising GF of particular partition ideals. We generalize ideals of ordered graphs to ordered uniform hypergraphs and show two dichotomies for their GF. The first result is a constant to linear jump for k-uniform hypergraphs. The second result establishes the polynomial to exponential jump for 3-uniform hypergraphs. That is, there are no ordered hypergraph ideals with GF strictly inside the constant-linear and polynomial- exponential range. We obtain in both dichotomies tight upper bounds. Finally, in a quite general setting we present several methods how to generate for various combinatorial structures pairs of sets defining two ideals with iden- tical GF. We call these pairs Wilf equivalent pairs and use the automorphism method and the replacement method to obtain such pairs. 1
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Various Approaches to Szroeter’s Test for Regression Quantiles
Kalina, Jan ; Peštová, B.
Regression quantiles represent an important tool for regression analysis popular in econometric applications, for example for the task of detecting heteroscedasticity in the data. Nevertheless, they need to be accompanied by diagnostic tools for verifying their assumptions. The paper is devoted to heteroscedasticity testing for regression quantiles, while their most important special case is commonly denoted as the regression median. Szroeter’s test, which is one of available heteroscedasticity tests for the least squares, is modified here for the regression median in three different ways: (1) asymptotic test based on the asymptotic representation for regression quantiles, (2) permutation test based on residuals, and (3) exact approximate test, which has a permutation character and represents an approximation to an exact test. All three approaches can be computed in a straightforward way and their principles can be extended also to other heteroscedasticity tests. The theoretical results are expected to be extended to other regression quantiles and mainly to multivariate quantiles.
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Various Approaches to Szroeter’s Test for Regression Quantiles
Kalina, Jan ; Peštová, Barbora
Regression quantiles represent an important tool for regression analysis popular in econometric applications, for example for the task of detecting heteroscedasticity in the data. Nevertheless, they need to be accompanied by diagnostic tools for verifying their assumptions. The paper is devoted to heteroscedasticity testing for regression quantiles, while their most important special case is commonly denoted as the regression median. Szroeter’s test, which is one of available heteroscedasticity tests for the least squares, is modified here for the regression median in three different ways: (1) asymptotic test based on the asymptotic representation for regression quantiles, (2) permutation test based on residuals, and (3) exact approximate test, which has a permutation character and represents an approximation to an exact test. All three approaches can be computed in a straightforward way and their principles can be extended also to other heteroscedasticity tests. The theoretical results are expected to be extended to other regression quantiles and mainly to multivariate quantiles.
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Weighted Halfspace Depths and Their Properties
Kotík, Lukáš
Statistical depth functions became well known nonparametric tool of multivariate data analyses. The most known depth functions include the halfspace depth. Although the halfspace depth has many desirable properties, some of its properties may lead to biased and misleading results especially when data are not elliptically symmetric. The thesis introduces 2 new classes of the depth functions. Both classes generalize the halfspace depth. They keep some of its properties and since they more respect the geometric structure of data they usually lead to better results when we deal with non-elliptically symmetric, multimodal or mixed distributions. The idea presented in the thesis is based on replacing the indicator of a halfspace by more general weight function. This provides us with a continuum, especially if conic-section weight functions are used, between a local view of data (e.g. kernel density estimate) and a global view of data as is e.g. provided by the halfspace depth. The rate of localization is determined by the choice of the weight functions and theirs parameters. Properties including the uniform strong consistency of the proposed depth functions are proved in the thesis. Limit distribution is also discussed together with some other data depth related topics (regression depth, functional data depth)...
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Weighted Halfspace Depths and Their Properties
Kotík, Lukáš ; Hlubinka, Daniel (advisor) ; Omelka, Marek (referee) ; Mosler, Karl (referee)
Statistical depth functions became well known nonparametric tool of multivariate data analyses. The most known depth functions include the halfspace depth. Although the halfspace depth has many desirable properties, some of its properties may lead to biased and misleading results especially when data are not elliptically symmetric. The thesis introduces 2 new classes of the depth functions. Both classes generalize the halfspace depth. They keep some of its properties and since they more respect the geometric structure of data they usually lead to better results when we deal with non-elliptically symmetric, multimodal or mixed distributions. The idea presented in the thesis is based on replacing the indicator of a halfspace by more general weight function. This provides us with a continuum, especially if conic-section weight functions are used, between a local view of data (e.g. kernel density estimate) and a global view of data as is e.g. provided by the halfspace depth. The rate of localization is determined by the choice of the weight functions and theirs parameters. Properties including the uniform strong consistency of the proposed depth functions are proved in the thesis. Limit distribution is also discussed together with some other data depth related topics (regression depth, functional data depth)...
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Enumeration of compositions with forbidden patterns
Dodova, Borjana ; Klazar, Martin (advisor) ; Jelínek, Vít (referee)
Enumeration of pattern avoiding compositions of numbers Abstract The aim of this work was to find some new results for 3-regular compositions, i.e., for those compositions which avoid the set of patterns {121, 212, 11}. Those compositions can be regarded as a generalization of Carlitz composition. Based on the generating function of compositions avoiding the set of patterns {121, 11} and {212, 11} we derive an upper bound for the coefficients of the power series of the generating function of 3-regular compositions. Using the theory of finite automata we derive its lower bound. We develop this result further by defining 3-block compositions. For the generating function of 3-regular compositions we prove a recursive ralation. Besides that we also compute the generating function of compositions avoiding the set of patterns {312, 321} whose parts are in the set [d]. In the last section we prove that the generating function of Carlitz compositions is transcendental.
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