Essays in Statistical Learning for Finance

Financial data analysis often involves samples that are short relative to the questions of interest, asymmetric and leptokurtic distributions, and substantial estimator instability. In the chapters that follow, I address these difficulties by introducing new methods designed to provide reliable tools for empirical finance, while limiting reliance on unrealistic assumptions or on procedures that reduce variability at the cost of excessive bias. The second chapter presents SCGlasso, a sparse precision-matrix estimator for Gaussian graphical models in which the maximum number of edges is controlled through an ℓ0-constraint, while an ℓ1-penalty preserves numerical stability. The third chapter addresses the problem of portfolio selection by introducing an approach based on confidence sets which, rather than committing to a single empirical estimate, explicitly accounts for statistical uncertainty in the choice of the optimal portfolio. The fourth chapter proposes a covariance-matrix shrinkage rule whose regularization intensity is calibrated directly on portfolio volatility, rather than on a generic mean squared error criterion. The final chapter moves beyond the normality assumption underlying graphical models by combining a rank-based correlation estimator with the sparsity-constrained precision-estimation step, thereby accommodating the heavy-tailed marginals typical of financial returns while preserving control over the number of edges. Overall, this thesis contributes to the statistical analysis of financial data by developing methods that balance stability and interpretability in settings where standard assumptions are often violated. Across the different chapters, the common objective is not only to perform well by standard statistical criteria, but to provide procedures that are more closely connected to the structure of the financial problem under study.