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Title of the thesis: On Univariate Statistical Inference of Multidimensional Extremes

Thesis defender: Jaakko Pere
Opponent: Prof. Valentin Patilea, ENSAI, CREST, France
Custos: Prof. Pauliina Ilmonen, Aalto University School of Science

Extreme value theory (EVT) is a branch of statistics that studies the behavior of rare events. It provides many methods for quantifying risk and is applied in a wide range of fields, from finance to climatology. From the theoretical point of view, EVT focuses on the behavior of sample maxima. On the contrary, the behavior of sums or means of random variables can be described with the classical central limit theory. In philosophical terms, classical statistical models reflect the realm of moderation, balanced behavior captured by Aristotle's notion of the golden mean. On the other hand, EVT is interested in significant departures from the center of the distribution. Consequently, the mathematical models provided by the study of the extreme and mean behavior of a random variable have contrasting use cases. For example, an explosion of a powder keg may be triggered either by a sudden spark or by gradual heating. In the former case, EVT is more applicable, whereas the latter is better modeled with cumulative or average behavior.

While there is a comprehensive literature on univariate EVT, many risk management scenarios require an understanding of the joint behavior of multiple random variables. Consider, for example, a traffic light rating system for a wildfire risk when there is data about humidity, wind speed and temperature. However, the analysis of multidimensional extremes is challenging since there is no canonical ordering between multidimensional observations, such as vectors. Nevertheless, in some specific multidimensional settings, one can define a natural ordering. As a familiar example, consider archery. The locations of the arrows on the target can be described by two-dimensional vectors, which can be ordered by their distance from the bullseye.

The approach for multidimensional extremes in this thesis is to measure the extremity of an observation with a context-dependent metric. Then, univariate EVT is applied to the magnitudes of the observations given by the metric. The thesis provides alternatives to the methods motivated by the so-called multidimensional maximum domain of attraction condition that remains the touchstone of multidimensional EVT.

More precisely, practical statistical methods for measuring risk in terms of extreme quantiles are developed for hand-picked multivariate and infinite-dimensional settings. Various desired theoretical properties of the methods are stated and proved.

Thesis available for public display 10 days prior to the defence at .