Regarding quantifying uncertainties in prediction, two projects with different perspectives and application backgrounds are presented in this dissertation. The goal of the first project is to predict the corporate default risks based on large-scale time-to-event and covariate data in the context of controlling credit risks. Specifically, we propose a competing risks model to incorporate exits of companies due to default and other reasons. Because of the stochastic and dynamic nature of the corporate risks, we incorporate both company-level and market-level covariate processes into the event intensities. We propose a parsimonious Markovian time series model and a dynamic factor model (DFM) to efficiently capture the mean and correlation structure of the high-dimensional covariate dynamics. For estimating parameters in the DFM, we derive an expectation maximization (EM) algorithm in explicit forms under necessary constraints. For multi-period default risks, we consider both the corporate-level and the market-level predictions. We also develop prediction interval (PI) procedures that synthetically take uncertainties in the future observation, parameter estimation, and the future covariate processes into account. In the second project, to quantify the uncertainties in the maximum likelihood (ML) estimators and compute the exact tolerance interval (TI) factors regarding the nominal confidence level, we propose algorithms for two-sided control-the-center and control-both-tails TI for complete or Type II censored data following the (log)-location-scale family of distributions. Our approaches are based on pivotal properties of ML estimators of parameters for the (log)-location-scale family and utilize the Monte-Carlo simulations. While for Type I censored data, only approximate pivotal quantities exist. An adjusted procedure is developed to compute the approximate factors. The observed CP is shown to be asymptotically accurate by our simulation study. Our proposed methods are illustrated using real-data examples. / Ph. D.

Diese Arbeit stellt Modelle und Methoden vor, die für robuste Statistiken und ihre Anwendungen in verschiedenen Bereichen entwickelt wurden. Kapitel 2 stellt einen neuartigen Partitionierungs-Clustering-Algorithmus vor, der auf Expectiles basiert. Der Algorithmus bildet Cluster, die sich an das Endverhalten der Clusterverteilungen anpassen und sie dadurch robuster machen. Das Kapitel stellt feste Tau-Clustering- und adaptive Tau-Clustering-Schemata und ihre Anwendungen im Kryptowährungsmarkt und in der Bildsegmentierung vor. In Kapitel 3 wird ein faktorerweitertes dynamisches Modell vorgeschlagen, um das Tail-Verhalten hochdimensionaler Zeitreihen zu analysieren. Dieses Modell extrahiert latente Faktoren, die durch Extremereignisse verursacht werden, und untersucht ihre Wechselwirkung mit makroökonomischen Variablen mithilfe des VAR-Modells. Diese Methodik ermöglicht Impuls-Antwort-Analysen, Out-of-Sample-Vorhersagen und die Untersuchung von Netzwerkeffekten. Die empirische Studie stellt den signifikanten Einfluss von durch finanzielle Extremereignisse bedingten Faktoren auf makroökonomische Variablen während verschiedener Wirtschaftsperioden dar. Kapitel 4 ist eine Pilotanalyse zu Non Fungible Tokens (NFTs), insbesondere CryptoPunks. Der Autor untersucht die Clusterbildung zwischen digitalen Assets mithilfe verschiedener Visualisierungstechniken. Die durch CNN- und UMAP-Regression identifizierten Cluster werden mit Preisen und Merkmalen von CryptoPunks in Verbindung gebracht. Kapitel 5 stellt die Konstruktion eines Preisindex namens Digital Art Index (DAI) für den NFT-Kunstmarkt vor. Der Index wird mithilfe hedonischer Regression in Kombination mit robusten Schätzern für die Top-10-Liquid-NFT-Kunstsammlungen erstellt. Es schlägt innovative Verfahren vor, nämlich Huberisierung und DCS-t-Filterung, um abweichende Preisbeobachtungen zu verarbeiten und einen robusten Index zu erstellen. Darüber hinaus werden Preisdeterminanten des NFT-Marktes analysiert. / This thesis provides models and methodologies developed on robust statistics and their applications in various domains. Chapter 2 presents a novel partitioning clustering algorithm based on expectiles. The algorithm forms clusters that adapt to the tail behavior of the cluster distributions, making them more robust. The chapter introduces fixed tau-clustering and adaptive tau-clustering schemes and their applications in crypto-currency market and image segmentation. In Chapter 3 a factor augmented dynamic model is proposed to analyse tail behavior of high-dimensional time series. This model extracts latent factors driven by tail events and examines their interaction with macroeconomic variables using VAR model. This methodology enables impulse-response analysis, out-of-sample predictions, and the study of network effects. The empirical study presents significant impact of financial tail event driven factors on macroeconomic variables during different economic periods. Chapter 4 is a pilot analysis on Non Fungible Tokens (NFTs) specifically CryptoPunks. The author investigates clustering among digital assets using various visualization techniques. The clusters identified through regression CNN and UMAP are associated with prices and traits of CryptoPunks. Chapter 5 introduces the construction of a price index called the Digital Art Index (DAI) for the NFT art market. The index is created using hedonic regression combined with robust estimators on the top 10 liquid NFT art collections. It proposes innovative procedures, namely Huberization and DCS-t filtering, to handle outlying price observations and create a robust index. Furthermore, it analyzes price determinants of the NFT market.