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|Title:||Extreme Value Statistics: Second-Order Models and Applications to Metal Fatigue|
|Abstract:||When we are interested in information about the extreme tails of a distribution, classical statistical tools cannot be applied. To this end extreme value methods were constructed. The generalized Pareto distribution (GPD) is probably the most popular model for inference on the tail of a distribution. More specific the GPD is known to model the excesses over a certain threshold well. However, for the GPD to fit such excesses well, the threshold should often be rather large, thereby the model has to be restricted to only a small upper fraction of the data. To overcome this problem we propose two extensions of the GPD. Both extensions are motivated by the second-order refinement of the underlying Pareto-type model. One is a single parameter extension and is only applicable on the group of the heavy tailed distributions, with a tail index >0. Not only can the extended model be fitted to a larger fraction of the data, the resulting maximum likelihood for the tail index is asymptotically unbiased. The second model is a two-parameter extension and can be applied on a larger class of distributions. Both models are tested and applied to real data sets. The second part of the thesis concerns the applications of these extreme value statistics in the field of material science, metal fatigue. Failure of metals, when subjected to mechanical forces, is often caused by the presence of impurities, inclusions or defects. At these points cracks occur and start growing until they cause failure. Studies have been made to predict the so called fatigue limit, or threshold for the stress under which the initial cracks will not grow. Metals subjected to stress below this threshold will not easily fail. The size of the original inclusions has been proven to be crucial for this threshold, the larger the inclusions the weaker the metal. In this research second-order models are applied in the aim to estimate the distribution of the large inclusions. The classical method for this estimation is based on two-dimensional cuts of the specimen which need to be transformed into three-dimensional data. This transformation can only be done under large assumptions. Steels also become cleaner and cleaner which makes it more difficult to find observations. Hence also the way of detection of the inclusions becomes more important. To this end a new technique, based on ultrasonic waves, has been constructed in the industry. Raw data using this method has been collected. As a first part of the research a way to clean this data has constructed. The resulted data is then applied to estimate the largest inclusions in a piece of steel using the proposed extreme value techniques.|
|JRC Institute:||Institute for the Protection and Security of the Citizen|
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