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1

Bootstrap confidence regions based on the Mahalanobis depth measure of two-dimensional samples

100%

Kobylińska M., Wagner W., University of Warmia-and-Mazury in Olsztyn, Higher School of Information Technology and Management in Rzeszów

Acta Universitatis Lodziensis. Folia Oeconomica

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2008

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vol. 216

EN

Construction of confidence regions for multi-dimensional samples is usually performed with a known stochastic distribution of a random vector in question. However, for multidimensional studies of socio-economic phenomena, such an assumption is difficult to make. Bootstrap methods can be helpful. The main problem with its application is the aligning of respective vectors. To this end, depth measures are used which express the vector distance from the central vector system cluster. Among many such depth measures, the Mahalanobis measure is one of the easiest from a numerical point of view. This paper presents a bootstrap region creation algorithm. It was illustrated for a two-dimensional sample.

PL

W pracy przedstawiony został algorytm tworzenia obszarów bootstrapowych. Do konstrukcji tych obszarów wykorzystano miary zanurzania obserwacji w próbie. Konstrukcję zaprezentowano dla przypadku dwuwymiarowego.

2

Comparison of Selected Criteria for Determination of the Measure of Depth of an Observation in a Two-dimensional Sample

88%

Kobylińska M., University of Warmia and Mazury O.

Acta Universitatis Lodziensis. Folia Oeconomica

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2006

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vol. 196

EN

The measure of observation depth in multidimensional samples, introduced into statistical practice by Tukey, has become a new tool for data analysis. It is a proposed method for determining multi-dimensional positional statistics, particularly in the analysis of non-typical data with outstanding observations. Applying a rule of depth helps to overcome the difficulties associated with sorting multidimensional observations. The notion of data depth has been intensively developed by many scholars, and, consequently, various criteria of the measurement of observation depth in a multidimensional samples may be found in literature. This paper contains a comparison of selected criteria of the measurement of observation depth in a two-dimensional case.

PL

W artykule dokonano porównania wybranych kryteriów wyznaczania miary zanurzania obserwacji w statystycznej próbie dwuwymiarowej. Wnioski dotyczące porównania tych kryteriów wyciągnięto na podstawie własnych badań empirycznych na próbach dwuwymiarowych.

3

Numerical Presentation of the Selected Statistical Notions Based on Tukey's Concept of Observation Depth in the Sample

75%

Wagner W., Kobylińska M., Wiesław Wagner - Academy of Physical Education P., Małgorzata Kobylińska - University of Warmia and Mazury O.

Acta Universitatis Lodziensis. Folia Oeconomica

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2003

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vol. 164

EN

The study provides presentation of selected statistical concepts based on data depth by Tukey. The concepts as: the rang of depth, the half-space convex, contour, simplicial depth breakdown points, position numerical measures, the trimmed mean depth, regression depth and the set of generally positive points were exemplified in a two-dimensional space of the dataset. There are also given numerical algorithms in some cases to indicate already mentioned concepts and to study their affined transformation. There are also given example for a one-dimensional space besides general description

PL

W pracy omówiono definicję zanurzenia punktu w próbie oraz wywodzące się z lej koncepcji pewne inne pojęcie statystyczne. Przedstawiono między innymi wskaźniki określające stopień zanurzenia oraz zaproponowano metody numerycznego ich wyznaczania.

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3 Acta Universitatis Lodziensis. Folia Oeconomica

3 Kobylińska M.

2 Wagner W.

1 Higher School of Information Technology and Management in Rzeszów

1 Małgorzata Kobylińska - University of Warmia and Mazury O.

1 University of Warmia and Mazury O.

1 University of Warmia-and-Mazury in Olsztyn

1 Wiesław Wagner - Academy of Physical Education P.

1 2008

1 2006

1 2003

Bootstrap confidence regions based on the Mahalanobis depth measure of two-dimensional samples

Comparison of Selected Criteria for Determination of the Measure of Depth of an Observation in a Two-dimensional Sample

Numerical Presentation of the Selected Statistical Notions Based on Tukey's Concept of Observation Depth in the Sample