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The importance of pivoting is well established in the context of nonparametric confidence regions. It ensures enhanced coverage accuracy. However, pivoting for directional data cannot be achieved simply by rescaling. A somewhat cumbersome pivotal method, which involves passing first into a space of higher dimension, has been developed by Fisher and Hall for samples of unit vectors. Although that method has some advantages over nonpivotal techniques, it does suffer from certain drawbacks-in particular, the operation of passing to a higher dimension. Here we suggest alternative pivotal approaches, the implementation of which does not require us to increase the intrinsic dimension of the data and which in practice seem to achieve greater coverage accuracy. These methods are of two types: new pivotal bootstrap techniques and techniques that exploit the `'implicit pivotalness'' of the empirical likelihood algorithm. Unlike the method proposed by Fisher and Hall. these methods are also applicable to axial data and lead to the first pivotal, small-sample nonparametric confidence methods for mean principal or polar axes.

The paper describes briefly the instances where the analysis of geological data has required the development of new statistical theory and methods. A new instance of this fruitful interaction of statistics and geology is then given. The theory of finite deformations is developed. Some of the classical and novel statistical problems which arise when objects embedded in a deformed rock are measured for strain estimation are discussed. (Author)

Obtaining valuable information from given data requires the use of appropriate methods of analysis. For example, if a certain variable of interest is assumed to depend on a (set of) covariate(s), this relationship can be described by regression models based on a set of observations. Once a well fitting model is found not only analysis of dependencies but also predictions of the variable of interest can be performed. Classical data models assume to know the underlying data generating mechanism up to a few unknown parameters which are estimated based on the given data. In contrast, regression trees and random forests follow an algorithm to fit a model in a purely data-driven way without requiring any model assumptions, i.e., the data generating process remains unknown. Over the last decade important progress has been made in the field of distributional modeling with flexible additive data models. However, until now these developments have not yet been fully integrated into the idea of regression trees and random forests. Therefore, the main goal of this dissertation is to provide the missing synthesis by introducing distributional trees and forests. On the one hand, this novel methodology allows for a full specification of a probability distribution in each node of the tree by modeling all distribution parameters, also accounting for possible censoring, truncation, tail behavior etc. This offers a wide range of information as not only expected values but for example also exceeding probabilities for a certain threshold or confidence intervals can be predicted. On the other hand, the tree-based models select covariates and their possible interactions automatically such that no model specifications are required in advance. Moreover, smooth effects of the explanatory variables can be approximated by an ensemble of trees combined to a more stable and regularized forest model, while non-linear and non-additive effects are detected by the splitting algorithm used to build a, Obtaining valuable information from given data requires the use of appropriate methods of analysis. For example, if a certain variable of interest is assumed to depend on a (set of) covariate(s), this relationship can be described by regression models based on a set of observations. Once a well fitting model is found not only analysis of dependencies but also predictions of the variable of interest can be performed. Classical data models assume to know the underlying data generating mechanism up to a few unknown parameters which are estimated based on the given data. In contrast, regression trees and random forests follow an algorithm to fit a model in a purely data-driven way without requiring any model assumptions, i.e., the data generating process remains unknown. Over the last decade important progress has been made in the field of distributional modeling with flexible additive data models. However, until now these developments have not yet been fully integrated into the idea of regression trees and random forests. Therefore, the main goal of this dissertation is to provide the missing synthesis by introducing distributional trees and forests. On the one hand, this novel methodology allows for a full specification of a probability distribution in each node of the tree by modeling all distribution parameters, also accounting for possible censoring, truncation, tail behavior etc. This offers a wide range of information as not only expected values but for example also exceeding probabilities for a certain threshold or confidence intervals can be predicted. On the other hand, the tree-based models select covariates and their possible interactions automatically such that no model specifications are required in advance. Moreover, smooth effects of the explanatory variables can be approximated by an, Lisa Schlosser, Kumulative Dissertation aus vier Artikeln, Dissertation University Innsbruck 2020

Accurate wind forecasts are of great importance for a wide range of areas, e.g., in the energy sector, in air traffic control, or for private leisure planning. Probabilistic predictions allow for a precise risk assessment and thus enable a well-founded decision making process. This dissertation develops different statistical approaches in the framework of distributional modeling to generate reliable probabilistic forecasts of wind speed and wind direction. The first presented article introduces the general concept of time-adaptive training schemes. Such schemes are typically required in the framework of distributional regression modeling to correctly post-process weather forecasts, including wind forecasts. In order to compare different time-adaptive training schemes in a uniform and simple setup, the study is carried out on post-processing near-surface air temperature forecasts across central Europe. Results indicate that approaches using data from multiple years for model training are often superior to approaches using the most recent days only. This is due to a more stable temporal evolution of the estimated coefficients, which leads to an overall improved performance; for the stations tested, this even holds in case of changes in the underlying ensemble prediction system which is employed for model training. Bridging the gap to probabilistic wind forecasting, the second paper presents a time-adaptive bivariate Gaussian regression model for the zonal and meridional wind components, employing EPS wind predictions as explanatory variables. The bivariate Gaussian process accounts for the circular characteristics of wind by utilizing both wind components and, therefore, information of wind speed and direction. In the third article, an alternative probabilistic modeling strategy is introduced by developing distributional regression trees and forests for a circular response, namely wind direction. Due to the highly volatile nature of wind direction, the tree-based dist, Accurate wind forecasts are of great importance for a wide range of areas, e.g., in the energy sector, in air traffic control, or for private leisure planning. Probabilistic predictions allow for a precise risk assessment and thus enable a well-founded decision making process. This dissertation develops different statistical approaches in the framework of distributional modeling to generate reliable probabilistic forecasts of wind speed and wind direction. The first presented article introduces the general concept of time-adaptive training schemes. Such schemes are typically required in the framework of distributional regression modeling to correctly post-process weather forecasts, including wind forecasts. In order to compare different time-adaptive training schemes in a uniform and simple setup, the study is carried out on post-processing near-surface air temperature forecasts across central Europe. Results indicate that approaches using data from multiple years for model training are often superior to approaches using the most recent days only. This is due to a more stable temporal evolution of the estimated coefficients, which leads to an overall improved performance; for the stations tested, this even holds in case of changes in the underlying ensemble prediction system which is employed for model training. Bridging the gap to probabilistic wind forecasting, the second paper presents a time-adaptive bivariate Gaussian regression model for the zonal and meridional wind components, employing EPS wind predictions as explanatory variables. The bivariate Gaussian process accounts for the circular characteristics of wind by utilizing both wind components and, therefore, information of wind speed and direction. In the third article, an alternative probabilistic modeling strategy is introduced by developing, Moritz N. Lang, Kumulative Dissertation aus drei Artikeln, Dissertation University of Innsbruck 2020

Accurate wind forecasts are of great importance for a wide range of areas, e.g., in the energy sector, in air traffic control, or for private leisure planning. Probabilistic predictions allow for a precise risk assessment and thus enable a well-founded decision making process. This dissertation develops different statistical approaches in the framework of distributional modeling to generate reliable probabilistic forecasts of wind speed and wind direction. The first presented article introduces the general concept of time-adaptive training schemes. Such schemes are typically required in the framework of distributional regression modeling to correctly post-process weather forecasts, including wind forecasts. In order to compare different time-adaptive training schemes in a uniform and simple setup, the study is carried out on post-processing near-surface air temperature forecasts across central Europe. Results indicate that approaches using data from multiple years for model training are often superior to approaches using the most recent days only. This is due to a more stable temporal evolution of the estimated coefficients, which leads to an overall improved performance; for the stations tested, this even holds in case of changes in the underlying ensemble prediction system which is employed for model training. Bridging the gap to probabilistic wind forecasting, the second paper presents a time-adaptive bivariate Gaussian regression model for the zonal and meridional wind components, employing EPS wind predictions as explanatory variables. The bivariate Gaussian process accounts for the circular characteristics of wind by utilizing both wind components and, therefore, information of wind speed and direction. In the third article, an alternative probabilistic modeling strategy is introduced by developing distributional regression trees and forests for a circular response, namely wind direction. Due to the highly volatile nature of wind direction, the tree-based dist, Accurate wind forecasts are of great importance for a wide range of areas, e.g., in the energy sector, in air traffic control, or for private leisure planning. Probabilistic predictions allow for a precise risk assessment and thus enable a well-founded decision making process. This dissertation develops different statistical approaches in the framework of distributional modeling to generate reliable probabilistic forecasts of wind speed and wind direction. The first presented article introduces the general concept of time-adaptive training schemes. Such schemes are typically required in the framework of distributional regression modeling to correctly post-process weather forecasts, including wind forecasts. In order to compare different time-adaptive training schemes in a uniform and simple setup, the study is carried out on post-processing near-surface air temperature forecasts across central Europe. Results indicate that approaches using data from multiple years for model training are often superior to approaches using the most recent days only. This is due to a more stable temporal evolution of the estimated coefficients, which leads to an overall improved performance; for the stations tested, this even holds in case of changes in the underlying ensemble prediction system which is employed for model training. Bridging the gap to probabilistic wind forecasting, the second paper presents a time-adaptive bivariate Gaussian regression model for the zonal and meridional wind components, employing EPS wind predictions as explanatory variables. The bivariate Gaussian process accounts for the circular characteristics of wind by utilizing both wind components and, therefore, information of wind speed and direction. In the third article, an alternative probabilistic modeling strategy is introduced by developing, Moritz N. Lang, Kumulative Dissertation aus drei Artikeln, Dissertation University of Innsbruck 2020

Obtaining valuable information from given data requires the use of appropriate methods of analysis. For example, if a certain variable of interest is assumed to depend on a (set of) covariate(s), this relationship can be described by regression models based on a set of observations. Once a well fitting model is found not only analysis of dependencies but also predictions of the variable of interest can be performed. Classical data models assume to know the underlying data generating mechanism up to a few unknown parameters which are estimated based on the given data. In contrast, regression trees and random forests follow an algorithm to fit a model in a purely data-driven way without requiring any model assumptions, i.e., the data generating process remains unknown. Over the last decade important progress has been made in the field of distributional modeling with flexible additive data models. However, until now these developments have not yet been fully integrated into the idea of regression trees and random forests. Therefore, the main goal of this dissertation is to provide the missing synthesis by introducing distributional trees and forests. On the one hand, this novel methodology allows for a full specification of a probability distribution in each node of the tree by modeling all distribution parameters, also accounting for possible censoring, truncation, tail behavior etc. This offers a wide range of information as not only expected values but for example also exceeding probabilities for a certain threshold or confidence intervals can be predicted. On the other hand, the tree-based models select covariates and their possible interactions automatically such that no model specifications are required in advance. Moreover, smooth effects of the explanatory variables can be approximated by an ensemble of trees combined to a more stable and regularized forest model, while non-linear and non-additive effects are detected by the splitting algorithm used to build a, Obtaining valuable information from given data requires the use of appropriate methods of analysis. For example, if a certain variable of interest is assumed to depend on a (set of) covariate(s), this relationship can be described by regression models based on a set of observations. Once a well fitting model is found not only analysis of dependencies but also predictions of the variable of interest can be performed. Classical data models assume to know the underlying data generating mechanism up to a few unknown parameters which are estimated based on the given data. In contrast, regression trees and random forests follow an algorithm to fit a model in a purely data-driven way without requiring any model assumptions, i.e., the data generating process remains unknown. Over the last decade important progress has been made in the field of distributional modeling with flexible additive data models. However, until now these developments have not yet been fully integrated into the idea of regression trees and random forests. Therefore, the main goal of this dissertation is to provide the missing synthesis by introducing distributional trees and forests. On the one hand, this novel methodology allows for a full specification of a probability distribution in each node of the tree by modeling all distribution parameters, also accounting for possible censoring, truncation, tail behavior etc. This offers a wide range of information as not only expected values but for example also exceeding probabilities for a certain threshold or confidence intervals can be predicted. On the other hand, the tree-based models select covariates and their possible interactions automatically such that no model specifications are required in advance. Moreover, smooth effects of the explanatory variables can be approximated by an, Lisa Schlosser, Kumulative Dissertation aus vier Artikeln, Dissertation University Innsbruck 2020