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Depth of variance matrices
Brabenec, Tomáš ; Nagy, Stanislav (advisor) ; Hlubinka, Daniel (referee)
The scatter halfspace depth is a quite recently established concept which extends the idea of the location halfspace depth for positive definite matrices. It provides an interest- ing insight into the problem of suitability quantification of a matrix for the description of the covariance structure of the multivariate distribution. The thesis focuses on the investigation of theoretical properties of the depth for both general and more specific probability distributions which can be used for data analysis. It turns out that the es- timators of scatter parameters based on the empirical scatter depth are quite effective even under relatively weak assumptions. These estimators are useful especially for dealing with a sample containing outliers or contaminating observations. 1
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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
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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