UC Davis

While statistical modeling of distributional data has gained increased attention, the case of multivariate distributions has been somewhat neglected despite its relevance in various applications. This is because distributions do not form a vector space and the Wasserstein distance that is commonly used in distributional data analysis poses challenges for multivariate distributions. This dissertation proposes two statistical frameworks including distributional regression models with multivariate distributions as responses paired with Euclidean vector predictors, and representations of multivariate distributions in the linear space. We propose distributional regression models with multivariate distributions as responses paired with Euclidean vector predictors, working with the sliced Wasserstein distance, which is based on a slicing transform from the multivariate distribution space to the sliced distribution space. We introduce two regression approaches: one that directly incorporates the sliced Wasserstein distance in the multivariate distribution space, and a second approach that employs a univariate distribution regression for each slice. We develop both global and local Fréchet regression methods for these approaches and establish asymptotic convergence for sample-based estimators. The proposed regression methods are illustrated in simulations and through applications to joint distributions of excess winter death rates and winter temperature anomalies in European countries as a function of base winter temperature, as well as two applications in finance. For the representations of multivariate distributions, we discuss a transformation approach to map multivariate distributions to a Hilbert space. Then tools from functional data analysis including functional principal component analysis, functional modes of variation and functional regression can be adopted in the Hilbert space. Convergence results of Hilbert space representations can then be obtained for a general class of such transformations. The proposed method is illustrated by modeling joint distributions of systolic and diastolic blood pressure data for samples of individuals as well as joint distributions of a sample of bivariate temperature data in the United States.