Leveraging Data Exchangeability for a More Reliable and Interpretable Functional Data Analysis

Most statistical methods heavily rely on assumptions about the specific data distribution family (i.e., parametric Statistics). In this work, I highlight that while these assumptions can enhance interpretability and strengthen statistical analysis from a mathematical, computational, and purely statistical perspective, they may also lead to unreliable statistical conclusions in modern contexts such as functional data analysis. As an alternative, I propose an approach that, rather than relying on the specific distribution of the data, leverages the exchangeability property induced by the i.i.d. assumption. Permutation tests are an example of this solution in the framework of functional hypothesis tests and conformal prediction an example in the context of probabilistic prediction of functional data.