Empirical Likelihood in Functional Data Analysis

Functional data analysis (FDA) studies data that include infinite-dimensional functions or objects, generalizing traditional univariate or multivariate observations from each study unit. Among inferential approaches without parametric assumptions, empirical likelihood (EL) offers a principled method in that it extends the framework of parametric likelihood ratio–based inference via the nonparametric likelihood. There has been increasing use of EL in FDA due to its many favorable properties, including self-normalization and the data-driven shape of confidence regions. This article presents a review of EL approaches in FDA, starting with finite-dimensional features, then covering infinite-dimensional features. We contrast smooth and nonsmooth frameworks in FDA and show how EL has been incorporated into both of them. The article concludes with a discussion of some future research directions, including the possibility of applying EL to conformal inference.