New Topics in Nonlinear Functional Data Analysis

Functional data analysis (FDA) comprises statistical methods for data that can be considered as partial or full observations of random curves, surfaces, or related smooth objects. Although much progress has been made in the last decades, a majority of these methods is linear and many important techniques remain absent from this infinite-dimensional branch of statistics. In this dissertation, two key concepts from nonlinear multivariate statistics are introduced to the FDA toolbox: copulas and power variations. Both offer entirely new nonparametric ways to analyse dependence structures of various infinite-dimensional random objects and have immediate applications in fields such as mathematical finance or physics. Due to the intrinsic infinite dimensionality, however, there are significant differences to their corresponding multivariate counterparts which give rise to various mathematical challenges that are addressed in this work