Contributions in functional data analysis and functional-analytic statistics

Functional data analysis is the study of statistical algorithms which are applied in the scenario when the observed data is a collection of functions. Since this type of data is becoming cheaper and easier to collect, there is an increased need to develop statistical tools to handle such data. The first part of this thesis focuses on deriving distances between distributions over function spaces and applying these to two-sample testing, goodness-of-fit testing and sample quality assessment. This presents a wide range of contributions since currently there exists either very few or no methods at all to tackle these problems for functional data. The second part of this thesis adopts the functional-analytic perspective to two statistical algorithms. This is a perspective where functions are viewed as living in specific function spaces and the tool box of functional analysis is applied to identify and prove properties of the algorithms. The two algorithms are variational Gaussian processes, used widely throughout machine learning for function modelling with large observation data sets, and functional statistical depth, used widely as a means to evaluate outliers and perform testing for functional data sets. The results presented contribute a taxonomy of the variational Gaussian process methodology and multiple new results in the theory of functional depth including the open problem of providing a depth which characterises distributions on function spaces.