Dimension Reduction of High-Dimensional Extremes using Information Criteria

The dependence structure of extremes for multivariate regularly varying random vectors of dimension d is characterized by the angular measure. Estimating the angular measure is challenging in high dimensions due to its complexity. In this thesis, we derive information criteria to reduce the dimension of high-dimensional extremes, investigating both the case where the dimension d is fixed and the case where the dimension d tends to infinity. The first approach is based on the concept of sparse regular variation introduced by Meyer and Wintenberger (2021). In this framework, we derive information criteria to select the number of directions in which extreme events occur, such as a Bayesian information criterion (BIC), a mean-squared error-based information criterion (MSEIC), and a quasi-Akaike information criterion (QAIC) based on the Gaussian likelihood function. For fixed dimension we prove that the AIC of Meyer and Wintenberger (2023) and the MSEIC are inconsistent information criteria for the number of extreme directions, whereas the BIC and the QAIC are consistent information criteria. In high-dimensions we derive for all information criteria sufficient conditions for consistency. ... mehrFor the second approach, we use Principal Component Analysis (PCA) for dimension reduction and estimate the number of significant principal components of the empirical covariance matrix of the angular measure under the assumption of a spiked covariance structure. To this end, we develop Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) to estimate the location of the spiked eigenvalue of the covariance matrix, reflecting the number of significant components, and investigate the consistency of these information criteria. In the high-dimensional setting, using techniques from random matrix theory, we derive sufficient conditions for the AIC and the BIC to be consistent. Finally, the performance of the information criteria is evaluated in a simulation study and applied to wind speed data and high-dimensional precipitation data.