Multivariate functional data often present theoretical and practical complications which are not found in their univariate functional counterparts. One of these is a situation where the component functions of multivariate functional data are subject to mutual time warping. That is, the component processes exhibit a similar shape but are subject to systematic phase variation across their time domains. This dissertation addresses this previously unconsidered mode of warping with the introduction of multivariate time warping models which rely on either time-shifting or nonlinear time-distortion frameworks. In the first chapter, we introduce a shift-warping model for multivariate time relations. This model is motivated by the Zürich longitudinal growth dataset, in which the growth trajectories for multiple body parts were observed from birth to adulthood for a sample of children. The proposed method differs from existing registration methods for functional data in a fundamental way. Namely, instead of focusing on the traditional approach to warping, where one aims to recover individual-specific registration, this technique focuses on shift registration across the components of a multivariate functional data vector on a population-wide level. After applying the method to the Zürich data, we find that there exists an archetypal ordering of pubertal growth spurts across modalities: on average, legs tend to experience peak growth velocity approximately a half a year before standing height and arms, which themselves tend to precede the growth spurt of the spine by half a year. Our proposed estimates for these shifts are identifiable, enjoy parametric rates of convergence, and often have intuitive physical interpretations, all in contrast to traditional curve-specific registration approaches. Finite sample properties of these estimators were also investigated in simulation studies.

The second chapter of this dissertation introduces the Latent Transport Model for multivariate functional data. This model widens the class of possible cross-component warps from simple shifts to flexible and nonlinear transport functions. The proposed approach combines a random amplitude factor for each component with population-based registration across the components of a multivariate functional data vector. It also includes a latent population function, which corresponds to a common underlying trajectory as well as a subject-specific warping component. This model allows for meaningful interpretation and is well suited to represent functional vector data. We also propose estimators for all components of the model, which not only lead to a novel representation for multivariate functional data, but can also be used in downstream analyses like Fréchet regression. Rates of convergence are established when curves are fully observed or observed with measurement error. The usefulness of the model, interpretations, and practical aspects are illustrated in simulations and with application to multivariate human growth curves as well as multivariate environmental pollution data.

Regression analysis, a cornerstone of statistical methodology, confronts emerging challenges in the realm of complex, non-Euclidean data. As responses are more often found in a general metric space that lacks a vector space structure, conventional regression techniques tailored for Euclidean spaces face a substantial hurdle. This dissertation strategically addresses this critical gap by formulating efficient and theoretically rigorous regression techniques specifically designed for two prominent non-Euclidean data types: networks and probability measures.

By viewing each network as an individual data point that lies in the space of graph Laplacians, we introduce a unifying intrinsic framework for network-response regression. The proposed regression models extend both linear and local linear regression to network responses, taking the form of weighted Fréchet means. The practical utility and effectiveness of this framework are demonstrated through simulations and real-world data applications.

Moving beyond network-response regression, the dissertation advances novel regression models tailored for probability measures on vector covariates, especially when sample sizes vary and are limited for some measures. Direct utilization of empirical measures sidesteps challenges associated with density estimation, enabling the proposed models to perform exceptionally well despite variable sample sizes and limited data.

Finally, we extend the proposed regression framework for probability measures to survival analysis. Specifically, we introduce an innovative regression model designed for right-censored survival data across diverse populations. This approach models the underlying probability measure and the corresponding nonparametric Kaplan-Meier estimator for each subgroup as elements within the Wasserstein space of probability measures. The demonstrated superior performance in finite sample scenarios positions it as an advanced alternative to the Cox proportional hazards model.

This dissertation explores statistical methodologies, particularly causal inference methods and Functional Data Analysis (FDA) methods, applied to problems in social and biomedical sciences. The work is divided into three distinct parts.

The first part studies the integration of external data in randomized controlled trials (RCTs), which is gaining interest from the FDA and industry, particularly when RCTs face limitations. The research has two main objectives: first, to introduce causal weighting estimators for improved statistical power while maintaining type I error rates. Second, we propose causal inference methods for estimating long-term treatment effects in the open-label extension phase of phase III studies, where subjects switch from placebo to the experimental treatment after the primary endpoint. The practical application of these methods is demonstrated through a phase III clinical trial in rare disease. The second part studies the U.S. Supreme Court dynamics by analyzing the temporal evolution of the underlying policy positions of the Supreme Court Justices as reflected by their actual voting data, using functional data analysis methods. The flexible nonparametric method makes it possible to represent highly complex dynamic trajectories by a few principal components and dissect the time-dynamics of policy positions at the level of individual Justices, as well as providing a comprehensive view of the ideology evolution over the history of Supreme Court since its establishment. In addition to quantifying individual Justice's policy positions, we uncover average changes over time and also the major patterns of change over time.

The third part studies gerrymandering. While the Court has been hesitant to intervene in partisan gerrymandering, it has been more willing to address racial gerrymandering due to the clear constitutional protections against racial discrimination in voting. Legislators engaged in partisan gerrymandering may use race as a proxy, targeting racial voting blocs to either concentrate or dilute their power. This can lead to situations where partisan gerrymandering inadvertently results in racial gerrymandering, creating a loophole in litigation. First, we introduce a novel metric of representation weight to measure individual voter representation in redistricting, filling a gap in existing research. Second, we estimate the quantile treatment effect to quantify the distributional impact of partisan redistricting on representation weight and the racial disparities in those effects. The utility of our framework is demonstrated through analyses of current redistricting litigations in Alabama, Florida, and Texas.

Data in the form of temporal point processes has received increasing attention in the literature due to advances in technology that allow to record and store large amounts of information such as the individual random times at which an event of interest occurs. These processes describe natural random phenomena that arise in the form of event time arrivals over a time window of interest, and provide a natural modeling framework for complex data such as bike pickup arrivals at a bike station, COVID-19 cases in a country, earthquake aftershocks, arrivals of phone calls at a call center, among many others, where both the number of events along with the event times themselves are random. This type of data can then be naturally seen in the form of samples of temporal point processes, which has also been termed replicated point process data, as here the random phenomena is repeatedly generated like for example over each day in the bike pickups point process or across each country in the case of COVID-19 cases. When further information in the form of Euclidean vector covariates is coupled with the temporal point process, statistical methodology such as regression methods becomes of paramount importance to analyze and explore point process data. It is of great interest to see how such covariates are associated with these complex random objects.

The first chapter is devoted to the development of a fully non-parametric regression method for temporal Cox point processes as responses with Euclidean vectors as predictors (Gajardo and Müller, 2022), which is based on the recently proposed Fréchet regression framework. Cox point processes are of central importance in the point process theory as they are natural generalizations of the well known non-homogeneous Poisson process but where the intensity function is allowed to be random itself. The latter property allows to model replicated point process data as each replication then comes from an independent and identically distributed underlying intensity function and thus gives more flexibility for modelling repeated random phenomena. We derive theoretical convergence rates for the proposed regression function in an increasing in-fill intensity asymptotic framework, where individual intensities are allowed to suitably diverge with sample size and allows to consistently recover the density function where the event arrivals come from as well as the conditional intensity factor up to a common constant.

The statistical analysis of complex objects such as point processes can be seen in a more general framework of random objects taking values in a metric space. Often such objects are infinite-dimensional in nature and the underlying metric space may not be a linear vector space, which leads to challenges for statistical analysis. A useful modeling assumption is that such random objects lie in a low-dimensional manifold so that low-dimensional representations are sufficient to describe and analyze such objects; this framework is often referred to as manifold learning in the literature. In the second chapter, we develop an inverse map from the Euclidean low-dimensional representation of the random objects that is obtained by adopting the well known ISOMAP procedure along with Multidimensional Scaling (MDS) back to object space, where we assume the manifold is isometric to a subset of a low-dimensional Euclidean space. For this, we employ the recently proposed framework of Fréchet regression and derive rates of convergence for the proposed inverse map. For the case of probability distributions in 2-Wasserstein space as random objects, we show that the proposed inverse map allows interpretation of the effect that each MDS component has on the distributional data and moreover derive convergence results for the case when only an increasing sample of observations coming from each probability distribution is available.

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.