Modern large data sets inevitably contain outliers that deviate from the model assumptions. However, many widely used estimators, such as maximum likelihood estimators and least squared estimators, perform weakly with the existence of outliers. Alternatively, many statistical modeling approaches have matrices as the parameters. We consider penalized estimators for matrix-valued parameters with a focus on their robustness properties in the presence of outliers. We propose a general framework for robust modeling with matrix-valued parameters by minimizing robust loss functions with penalization. However, there are challenges to this approach in both computation and theoretical analysis. To tackle the computational challenges from the large size of the data, non-smoothness of robust loss functions, and the slow speed of matrix operations, we propose to apply the Frank-Wolfe algorithm, a first-order algorithm for optimization on a restricted region with low computation burden per iteration. Theoretically, we establish finite-sample error bounds under high-dimensional settings. We show that the estimation errors are bounded by small terms and converge in probability to zero under mild conditions in a neighborhood of the true model. Our method accommodates a broad classes of modeling problems using robust loss functions with penalization. Concretely, we study three cases: matrix completion, multivariate regression, and network estimation. For all cases, we illustrate the robustness of the proposed method both theoretically and numerically. / Statistics

In this thesis we study the types of algebraically closed valued fields (ACVF). We prove the definable types of ACVF are residual and valuational and provide a defining schema for the definable types. We then conclude that all the types are invariant. / Thesis / Master of Science (MSc)

This dissertation proposes new statistical methods for modeling and inference for two specific types of time series: integer-valued data and multivariate nonstationary extreme data. We rely on the class of integer-valued autoregressive (INAR) processes for the former, proposing a novel, flexible and elegant way of modeling count phenomena. As for the latter, we are interested in the human brain and its multi-channel electroencephalogram (EEG) recordings, a natural source of extreme events. Thus, we develop new extreme value theory methods for analyzing such data, whether in modeling the conditional extremal dependence for brain connectivity or clustering extreme brain communities of EEG channels. Regarding integer-valued time series, INAR processes are generally defined by specifying the thinning operator and either the innovations or the marginal distributions. The major limitations of such processes include difficulties deriving the marginal properties and justifying the choice of the thinning operator. To overcome these drawbacks, this dissertation proposes a novel approach for building an INAR model that offers the flexibility to prespecify both marginal and innovation distributions. Thus, the thinning operator is no longer subjectively selected but is rather a direct consequence of the marginal and innovation distributions specified by the modeler. Novel INAR processes are introduced following this perspective; these processes include a model with geometric marginal and innovation distributions (Geo-INAR) and models with bounded innovations. We explore the Geo-INAR model, which is a natural alternative to the classical Poisson INAR model. The Geo-INAR process has interesting stochastic properties, such as MA($\infty$) representation, time reversibility, and closed forms for the $h$-th-order transition probabilities, which enables a natural framework to perform coherent forecasting. In the front of multivariate nonstationary extreme data, the focus lies on multi-channel epilepsy data. Epilepsy is a chronic neurological disorder affecting more than 50 million people globally. An epileptic seizure acts like a temporary shock to the neuronal system, disrupting normal electrical activity in the brain. Epilepsy is frequently diagnosed with EEGs. Current statistical approaches for analyzing EEGs use spectral and coherence analysis, which do not focus on extreme behavior in EEGs (such as bursts in amplitude), neglecting that neuronal oscillations exhibit non-Gaussian heavy-tailed probability distributions. To overcome this limitation, this dissertation proposes new approaches to characterize brain connectivity based on extremal features of EEG signals. Two extreme-valued methods to study alterations in the brain network are proposed. One method is Conex-Connect, a pioneering approach linking the extreme amplitudes of a reference EEG channel with the other channels in the brain network. The other method is Club Exco, which clusters multi-channel EEG data based on a spherical $k$-means procedure applied to the "pseudo-angles," derived from extreme amplitudes of EEG signals. Both methods provide new insights into how the brain network organizes itself during an extreme event, such as an epileptic seizure, in contrast to a baseline state.

The main objective of this thesis is to develop a martingale-type solution to optimal consumption--investment choice problems ([Merton, 1969] and [Merton, 1971]) under time-varying incomplete preferences driven by externalities such as patience, socialization effects, and market volatility. The market is composed of multiple risky assets and multiple consumption goods, while in addition there are multiple fluctuating preference parameters with inexact values connected to imprecise tastes. Utility maximization becomes a multi-criteria problem with possibly function-valued criteria. To come up with a complete characterization of the solutions, first we motivate and introduce a set-valued stochastic process for the dynamics of multi-utility indices and formulate the optimization problem in a topological vector space. Then, we modify a classical scalarization method allowing for infiniteness and randomness in dimensions and prove results of equivalence to the original problem. Illustrative examples are given to demonstrate practical interests and method applicability progressively. The link between the original problem and a dual problem is also discussed, relatively briefly. Finally, by using Malliavin calculus with stochastic geometry, we find optimal investment policies to be generally set-valued, each of whose selectors admits a four-way decomposition involving an additional indecisiveness risk-hedging portfolio. Our results touch on new directions for optimal consumption--investment choices in the presence of incomparability and time inconsistency, also signaling potentially testable assumptions on the variability of asset prices. Simulation techniques for set-valued processes are studied for how solved optimal policies can be computed in practice. / 2025-05-12T00:00:00Z

This work can be roughly divided into two parts. Initially, it may be considered a continuation of the very interesting research on the topic of Lattice-Valued Convergence Spaces given by Jager [2001, 2005]. The alternate axioms presented here seem to lead to theorems having proofs more closely related to standard arguments used in Convergence Space theory when the Lattice is L = f0; 1g:Various Subcategories are investigated. One such subconstruct is shown to be isomorphic to the category of Lattice Valued Fuzzy Convergence Spaces defined and studied by Jager [2001]. Our principal category is shown to be a topological universe and contains a subconstruct isomorphic to the category of probabilistic convergence spaces discussed in Kent and Richardson [1996] when L = [0; 1]: Fundamental work in lattice-valued convergence from the more general perspective of monads can be found in Gahler [1995]. Secondly, diagonal axioms are defned in the category whose objects consist of all the lattice valued convergence spaces. When the latter lattice is linearly ordered, a diagonal condition is given which characterizes those objects in the category that are determined by probabilistic convergence spaces which are topological. Certain background information regarding filters, convergence spaces, and diagonal axioms with its dual are given in Chapter 1. Chapter 2 describes Probabilistic Convergence and associated Diagonal axioms. Chapter 3 defines Jager convergence and proves that Jager's construct is isomorphic to a bireáective subconstruct of SL-CS. Furthermore, connections between the diagonal axioms discussed and those given by Gahler are explored. In Chapter 4, further categorical properties of SL-CS are discussed and in particular, it is shown that SL-CS is topological, cartesian closed, and extensional. Chapter 5 explores connections between diagonal axioms for objects in the sub construct δ(PCS) and SL-CS. Finally, recommendations for further research are provided.

This thesis presents innovative approaches for controlling the False Discovery Rate (FDR) in both high-dimensional statistical inference and finite-sample cases, addressing challenges arising from various dependency structures in the data. The first project introduces novel multiple testing methods for matrix-valued data, motivated by an electroencephalography (EEG) experiment, where we model the inherent complex row-column cross-dependency using a matrix normal distribution. We proposed two methods designed for structured matrix-valued data, to approximate the true FDP that captures the underlying cross-dependency with statistical accuracy. In the second project, we focus on simultaneous testing of multivariate normal means under diverse covariance matrix structures. By adjusting p-values using a BH-type step-up procedure tailored to the known correlation matrix, we achieve robust finite-sample FDR control. Both projects demonstrate superior performance through extensive numerical studies and real-data applications, significantly advancing the field of multiple testing under dependency. The third project presented exploratory simulation results to demonstrate the methods constructed based on the paired-p-values framework that controls the FDR within the multivariate normal means testing framework. / Statistics

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The present work is about measure-valued diffusion processes, which are aligned with two distinct geometries on the set of probability measures. In the first part we focus on a stochastic partial differential equation, the Dean-Kawasaki equation, which can be considered as a natural candidate for a Langevin equation on probability measures, when equipped with the Wasserstein distance. Apart from that, the dynamic in question appears frequently as a model for fluctuating density fields in non-equilibrium statistical mechanics. Yet, we prove that the Dean-Kawasaki equation admits a solution only in integer parameter regimes, in which case the solution is given by a particle system of finite size with mean field interaction. For the second part we restrict ourselves to positive probability measures on a finite set, which we identify with the open standard unit simplex. We show that Brownian motion on the simplex equipped with the Aitchison geometry, can be interpreted as a replicator dynamic in a white noise fitness landscape. We infer three approximation results for this Aitchison diffusion. Finally, invoking Fokker-Planck equations and Wasserstein contraction estimates, we study the long time behavior of the stochastic replicator equation, as an example of a non-gradient drift diffusion on the Aitchison simplex.

In this work we present a new tool for the convergence analysis of numerical optimization methods. It is based on the concepts of the Clarke derivative and set-valued mappings. Our goal is to apply this tool to minimization problems with non-smooth and noisy objective functions. After deriving a necessary condition for minimizers of such functions, we examine two unconstrained optimization routines. First, we prove new convergence theorems for Implicit Filtering and General Pattern Search. Then we show how these results can be used in practice, by executing some numerical computations. / Master of Science

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