Dortmund, Universität, Thesis (doctoral), 2001.

Zugl.: Hamburg, Helmut-Schmidt-Universiẗat, Diss., 2009.

Master-Arbeit Univ. St. Gallen, 2008.

Die vorliegende Diplomarbeit beschäftigt sich mit zwei Aspekten der statistischen Eigenschaften von Clusterverfahren. Zum einen geht die Arbeit auf die Frage der Existenz von unterschiedlichen Clusteranalysemethoden zur Strukturfindung und deren unterschiedlichen Vorgehensweisen ein. Die Methode des Abstandes zwischen Mannigfaltigkeiten und die K-means Methode liefern ausgehend von gleichen Daten unterschiedliche Endclusterungen. Der zweite Teil dieser Arbeit beschäftigt sich näher mit den asymptotischen Eigenschaften des K-means Verfahrens. Hierbei ist die Menge der optimalen Clusterzentren konsistent. Bei Vergrößerung des Stichprobenumfangs gegen Unendlich konvergiert diese in Wahrscheinlichkeit gegen die Menge der Clusterzentren, die das Varianzkriterium minimiert. Ebenfalls konvergiert die Menge der optimalen Clusterzentren für n gegen Unendlich gegen eine Normalverteilung. Es hat sich dabei ergeben, dass die einzelnen Clusterzentren voneinander abhängen. / The following thesis describes two different views onto the statistical characterics of clustering procedures. At first it adresses the questions whether different clustering methods exist to ascertain the structure of clusters and in what ays the strategies of these methods differ from each other. The method of distance between the manifolds as well as the k-means method provide different final clusters based on equal initial data. The second part of the thesis concentrates on asymptotic properties of the k-means procedure. Here the amount of optimal clustering centres is consistent. If the size of the sample range is enlarged towards infinity, it also converges in probability towards the amount of clustering centres which minimized the whithin cluster sum of squares. Likewise the amount of optimal clustering centres converges for infinity towards the normal distribution. The main result shows that the individual clustering centres are dependent on each other.

To extract the best possible information from geodetic and geophysical observations, it is necessary to select a model of the observation errors, mostly the family of Gaussian normal distributions. However, there are alternatives, typically chosen in the framework of robust M-estimation. We give a synopsis of well-known and less well-known models for observation errors and propose to select a model based on information criteria. In this contribution we compare the Akaike information criterion (AIC) and the Anderson Darling (AD) test and apply them to the test problem of fitting a straight line. The comparison is facilitated by a Monte Carlo approach. It turns out that the model selection by AIC has some advantages over the AD test.

To extract the best possible information from geodetic and geophysical observations, it is necessary to select a model of the observation errors, mostly the family of Gaussian normal distributions. However, there are alternatives, typically chosen in the framework of robust M-estimation. We give a synopsis of well-known and less well-known models for observation errors and propose to select a model based on information criteria. In this contribution we compare the Akaike information criterion (AIC) and the Anderson Darling (AD) test and apply them to the test problem of fitting a straight line. The comparison is facilitated by a Monte Carlo approach. It turns out that the model selection by AIC has some advantages over the AD test.

In this discusses four different approaches used during a statistics tutorial of a group of first year undergraduates studying computer science related degrees at the University of Westminster UK. The four approaches were each implemented in an attempt to keep the students interested in the statistics topics delivered. It was found that “Chalk and Talk” (i.e. board work) was not the best form of imparting knowledge to the students of the group as determined by student analysing feedback forms and generally observing student behaviour and listening to student comments over a number of years delivering statistics topics. The duration of each tutorial was two hours. The teaching strategies adopted were: a) A class quiz. b) Group explanation of material to members of the individual’s group. c) Group explanation of material to members of the entire class. d) Students teaching at the front of the class. Each of the methods will now be discussed with the relative merits and defects included for a comparison. It was found that each method worked better at the end of each module when the students were more familiar with the topics introduced on the module.

The generalized inverse Gaussian distribution has become quite popular in financial engineering. The most popular random variate generator is due to Dagpunar (1989). It is an acceptance-rejection algorithm method based on the Ratio-of-uniforms method. However, it is not uniformly fast as it has a prohibitive large rejection constant when the distribution is close to the gamma distribution. Recently some papers have discussed universal methods that are suitable for this distribution. However, these methods require an expensive setup and are therefore not suitable for the varying parameter case which occurs in, e.g., Gibbs sampling. In this paper we analyze the performance of Dagpunar's algorithm and combine it with a new rejection method which ensures a uniformly fast generator. As its setup is rather short it is in particular suitable for the varying parameter case. (authors' abstract) / Series: Research Report Series / Department of Statistics and Mathematics

In this discusses four different approaches used during a statistics tutorial of a group of first year undergraduates studying computer science related degrees at the University of Westminster UK. The four approaches were each implemented in an attempt to keep the students interested in the statistics topics delivered. It was found that “Chalk and Talk” (i.e. board work) was not the best form of imparting knowledge to the students of the group as determined by student analysing feedback forms and generally observing student behaviour and listening to student comments over a number of years delivering statistics topics. The duration of each tutorial was two hours. The teaching strategies adopted were: a) A class quiz. b) Group explanation of material to members of the individual’s group. c) Group explanation of material to members of the entire class. d) Students teaching at the front of the class. Each of the methods will now be discussed with the relative merits and defects included for a comparison. It was found that each method worked better at the end of each module when the students were more familiar with the topics introduced on the module.

Based on an application in the field of server consolidation, we consider the one-dimensional cutting stock problem with nondeterministic item lengths. After a short introduction to the general topic we investigate the case of normally distributed item lengths in more detail. Within this framework, we present two lower bounds as well as two heuristics to obtain upper bounds, where the latter are either based on a related (ordinary) cutting stock problem or an adaptation of the first fit decreasing heuristic to the given stochastical context. For these approximation techniques, dominance relations are discussed, and theoretical performance results are stated. As a main contribution, we develop a characterization of feasible patterns by means of one linear and one quadratic inequality. Based on this, we derive two exact modeling approaches for the nondeterministic cutting stock problem, and provide results of numerical simulations.