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.

中文翻译：

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函数数据分析中的实证似然


功能数据分析 （FDA） 研究包括无限维函数或对象的数据，概括来自每个研究单元的传统单变量或多变量观测值。在没有参数假设的推理方法中，经验似然 （EL） 提供了一种原则性方法，因为它通过非参数似然扩展了基于参数似然比的推理框架。由于 EL 具有许多有利特性，包括自归一化和数据驱动的置信区形状，因此 FDA 越来越多地使用 EL。本文回顾了 FDA 中的 EL 方法，从有限维特征开始，然后介绍无限维特征。我们对比了 FDA 中的平滑和非平滑框架，并展示了 EL 是如何被纳入两者的。本文最后讨论了一些未来的研究方向，包括将 EL 应用于共形推理的可能性。